[PAS] Probability Abstracts 103
Probability Abstract Service
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Sun May 4 16:03:13 CDT 2008
Probability Abstracts 103
This document contains abstracts 6753-6993 from
March-1-2008 to April-30-2008.
They have been mailed on May 4th, 2008.
This letter can be also found on line at
http://pas.imstat.org/Letters/letter_103.shtml
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6753. THE CONDITIONED RECONSTRUCTED PROCESS
Tanja Gernhard
We investigate a neutral model for speciation and extinction, the
constant
rate birth-death process. The process is conditioned to have $n$
extant species
today, we look at the tree distribution of the reconstructed trees--
i.e. the
trees without the extinct species. Whereas the tree shape distribution
is
well-known and actually the same as under the pure birth process, no
analytic
results for the speciation times were known. We provide the
distribution for
the speciation times and calculate the expectations analytically. This
characterizes the reconstructed trees completely. We will show how the
results
can be used to date phylogenies.
http://arxiv.org/abs/0803.0153
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6754. THE SQUARE NEGATIVE CORRELATION PROPERTY FOR GENERALIZED ORLICZ
BALLS
Jakub Onufry Wojtaszczyk
Antilla, Ball and Perissinaki proved that the squares of coordinate
functions
in $\ell_p^n$ are negatively correlated. This paper extends their
results to
balls in generalized Orlicz norms on R^n. From this, the concentration
of the
Euclidean norm and a form of the Central Limit Theorem for the
generalized
Orlicz balls is deduced. Also, a counterexample for the square negative
correlation hypothesis for 1-symmetric bodies is given.
Currently the CLT is known in full generality for convex bodies
(see the
paper "Power-law estimates for the central limit theorem for convex
sets" by B.
Klartag), while for generalized Orlicz balls a much more general
result is true
(see "The negative association property for the absolute values of
random
variables equidistributed on a generalized Orlicz ball" by M.
Pilipczuk and J.
O. Wojtaszczyk). While, however, both aforementioned papers are rather
long,
complicated and technical, this paper gives a simple and elementary
proof of,
eg., the Euclidean concentration for generalized Orlicz balls.
http://arxiv.org/abs/0803.0433
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6755. THE NEGATIVE ASSOCIATION PROPERTY FOR THE ABSOLUTE VALUES OF
RANDOM VARIABLES EQUIDISTRIBUTED ON A GENERALIZED ORLICZ BALL
Marcin Pilipczuk and Jakub Onufry Wojtaszczyk
Random variables equidistributed on convex bodies have received quite
a lot
of attention in the last few years. In this paper we prove the negative
association property (which generalizes the subindependence of
coordinate
slabs) for generalized Orlicz balls. This allows us to give a strong
concentration property, along with a few moment comparison
inequalities. Also,
the theory of negatively associated variables is being developed in
its own
right, which allows us to hope more results will be available.
Moreover, a
simpler proof of a more general result for $\ell_p^n$ balls is given.
http://arxiv.org/abs/0803.0434
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6756. INDIVIDUAL RISK AND LEBESGUE EXTENSION WITHOUT AGGREGATE
UNCERTAINTY
Yeneng Sun and Yongchao Zhang
Many economic models include random shocks imposed on a large number
(continuum) of economic agents with individual risk. In this context,
an exact
law of large numbers and its converse is presented in Sun [Journal of
Economic
Theory 126(2006), 31-69] to characterize the cancelation of individual
risk via
aggregation. However, it is well known that the Lebesgue unit interval
is not
suitable for modeling a continuum of agents in the particular setting.
The
purpose of this paper is to show that an extension of the Lebesgue unit
interval does work well as an agent space with various desirable
properties
associated with individual risk.
http://arxiv.org/abs/0803.0442
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6757. STEIN'S METHOD AND EXACT BERRY-ESS\'EEN ASYMPTOTICS FOR
FUNCTIONALS OF GAUSSIAN FIELDS
Ivan Nourdin (PMA) and Giovanni Peccati (LSTA)
We show how to detect optimal Berry-Ess\'een bounds in the normal
approximation of functionals of Gaussian fields. Our techniques are
based on a
combination of Malliavin calculus, Stein's method and the method of
moments and
cumulants, and provide de facto local (one term) Edgeworth expansions.
The
findings of the present paper represent a further refinement of the main
results proved in Nourdin and Peccati (2007b). Among several examples,
we
discuss three crucial applications: (i) to Toeplitz quadratic
functionals of
continuous-time stationary processes (extending results by Ginovyan
(1994) and
Ginovyan and Sahakyan (2007)), (ii) to ``exploding'' quadratic
functionals of a
Brownian sheet, and (iii) to a continuous-time version of the Breuer-
Major CLT
for functionals of a fractional Brownian motion.
http://arxiv.org/abs/0803.0458
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6758. ROUGH EVOLUTION EQUATIONS
Massimiliano Gubinelli and Samy Tindel
We show how to generalize Lyons' rough paths theory in order to give a
pathwise meaning to some nonlinear infinite-dimensional evolution
equations
associated to an analytic semigroup and driven by an irregular noise.
As an
illustration, we apply the theory to a class of 1d SPDEs driven by a
space-time
fractional Brownian motion.
http://arxiv.org/abs/0803.0552
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6759. SUPERPOSITION RULES AND STOCHASTIC LIE-SCHEFFERS SYSTEMS
Joan-Andreu L\'azaro-Cam\'i and Juan-Pablo Ortega
This paper proves a version for stochastic differential equations of the
Lie-Scheffers Theorem. This result characterizes the existence of
nonlinear
superposition rules for the general solution of those equations in
terms of the
involution properties of the distribution generated by the vector
fields that
define it. When stated in the particular case of standard deterministic
systems, our main theorem improves various aspects of the classical
Lie-Scheffers result. We show that the stochastic analog of the
classical
Lie-Scheffers systems can be reduced to the study of Lie group valued
stochastic Lie-Scheffers systems; those systems, as well as those
taking values
in homogeneous spaces are studied in detail. The developments of the
paper are
illustrated with several examples.
http://arxiv.org/abs/0803.0600
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6760. POLLING SYSTEMS WITH PARAMETER REGENERATION, THE GENERAL CASE
Iain MacPhee and Mikhail Menshikov and Dimitri Petritis and Serguei
Popov
We consider a polling model with multiple stations, each with
Poisson arrivals and a queue of infinite capacity. The service
regime is
exhaustive and there is Jacksonian feedback of served customers. What
is new
here is that when the server comes to a station it chooses the service
rate and
the feedback parameters at random; these remain valid during the whole
stay of
the server at that station. We give criteria for recurrence,
transience, and
existence of the $s$th moment of the return time to the empty state
for this
model. This paper generalizes the model when only two stations accept
arriving
jobs which was considered in \cite{MMPP}. Our results are stated in
terms of
Lyapunov exponents for random matrices. From the recurrence criteria
it can be
seen that the polling model with parameter regeneration can exhibit
the unusual
phenomenon of null recurrence over a thick region of parameter space.
http://arxiv.org/abs/0803.0625
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6761. THE THEORY OF FALLIBLE PROBABILITY AND THE DYNAMICS OF DEGREES
OF BELIEF
Amos Nathan
This monograph is an account of the theory of fallible probability and
of the
dynamics of degrees of belief. It discusses the first order subjective
theory
in which first order degrees of belief are expressed by subjective
probabilities and are updated by conditionalization (Bayes, 1764;
Ramsey,
1926), gives an improved exposition of the greater part of the
author's theory
of Probability Dynamics (Nathan, 2006) which should replace the so-
called
Probability Kinematics (Jeffrey, 1965), resolves the problem of New
Explanation
of Old Evidence (Jeffrey, 1995), provides a Theory of Confirmation,
and refutes
the Principle of Reflection (Van Fraassen, 1984).
http://arxiv.org/abs/0803.0630
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6762. A NOTE ON OPTIMAL PROBABILITY LOWER BOUNDS FOR CENTERED RANDOM
VARIABLES
Mark Veraar
In this note we obtain lower bounds for $\P(\xi\geq 0)$ and $
\P(\xi>0)$ under
assumptions on the moments of a centered random variable $\xi$. The
obtained
estimates are shown to be optimal and improve results from the
literature. The
results are applied to obtain probability lower bounds for second order
Rademacher chaos.
http://arxiv.org/abs/0803.0727
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6763. RECURRENCE OF THE TWISTED PLANAR RANDOM WALK
U. Haboeck
We show that the "twisted" planar random walk - which results by
summing up
stationary increments rotated by multiples of a fixed angle - is
recurrent
under diverse assumptions on the increment process. For example, if the
increment process is alpha-mixing and of finite second moment, then
the twisted
random walk is recurrent for every angle fixed choice of the angle out
of a set
of full Lebesgue measure, no matter how slow the mixing coefficients
decay.
http://arxiv.org/abs/0803.0724
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6764. TIME--SPACE HARMONIC POLYNOMIALS RELATIVE TO A L\'{E}VY PROCESS
Josep Llu\'is Sol\'e and Frederic Utzet
In this work, we give a closed form and a recurrence relation for a
family of
time--space harmonic polynomials relative to a L\'{e}vy process. We
also state
the relationship with the Kailath--Segall (orthogonal) polynomials
associated
to the process.
http://arxiv.org/abs/0803.0829
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6765. ON THE RUIN PROBLEM IN THE RENEWAL RISK PROCESSES PERTURBED BY
DIFFUSION
Min Song
In this paper, we consider the perturbed renewal risk process. Systems
of
integro-differential equations for the Gerber-Shiu functions at ruin
caused by
a claim and oscillation are established, respectively. The explicit
Laplase
transforms of Gerber-Shiu functions are obtained, while the closed form
expressions for the Gerber-Shiu functions are derived when the claim
amount
distribution is from the rational family. Finally, we present numerical
examples intended to illustrate the main results.
http://arxiv.org/abs/0803.0906
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6766. ON THE SECOND-ORDER CORRELATION FUNCTION OF THE CHARACTERISTIC
POLYNOMIAL OF A HERMITIAN WIGNER MATRIX
F. G\"otze and H. K\"osters
We consider the asymptotics of the second-order correlation function
of the
characteristic polynomial of a random matrix. We show that the known
result for
a random matrix from the Gaussian Unitary Ensemble essentially
continues to
hold for a general Hermitian Wigner matrix. Our proofs rely on an
explicit
formula for the exponential generating function of the second-order
correlation
function of the characteristic polynomial.
http://arxiv.org/abs/0803.0926
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6767. ON THE SECOND-ORDER CORRELATION FUNCTION OF THE CHARACTERISTIC
POLYNOMIAL OF A REAL SYMMETRIC WIGNER MATRIX
H. K\"osters
We consider the asymptotic behaviour of the second-order correlation
function
of the characteristic polynomial of a real symmetric random matrix.
Our main
result is that the existing result for a random matrix from the Gaussian
Orthogonal Ensemble essentially continues to hold for a general real
symmetric
Wigner matrix.
http://arxiv.org/abs/0803.0932
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6768. STATISTICAL ANALYSIS OF SELF-SIMILAR CONSERVATIVE FRAGMENTATION
CHAINS
Marc Hoffmann (LAMA) and Nathalie Krell (PMA)
We explore statistical inference in self-similar conservative
fragmentation
chains, when only (approximate) observations of the size of the
fragments below
a given threshold are available. This framework, introduced by Bertoin
and
Martinez, is motivated by mineral crushing in mining industry. The
underlying
estimated object is the step distribution of the random walk
associated to a
randomly tagged fragment that evolves along the genealogical tree
representation of the fragmentation process. We compute upper and
lower rates
of estimation in a parametric framework, and show that in the non-
parametric
case, the difficulty of the estimation is comparable to ill-posed linear
inverse problems of order 1 in signal denoising.
http://arxiv.org/abs/0803.0879
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6769. TOEPLITZ BLOCK MATRICES IN COMPRESSED SENSING
Florian Sebert and Leslie Ying and and Yi Ming Zou
Recent work in compressed sensing theory shows that $n\times N$
independent
and identically distributed (IID) sensing matrices whose entries are
drawn
independently from certain probability distributions guarantee exact
recovery
of a sparse signal with high probability even if $n\ll N$. Motivated
by signal
processing applications, random filtering with Toeplitz sensing
matrices whose
elements are drawn from the same distributions were considered and
shown to
also be sufficient to recover a sparse signal from reduced samples
exactly with
high probability. This paper considers Toeplitz block matrices as
sensing
matrices. They naturally arise in multichannel and multidimensional
filtering
applications and include Toeplitz matrices as special cases. It is
shown that
the probability of exact reconstruction is also high. Their
performance is
validated using simulations.
http://arxiv.org/abs/0803.0755
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6770. RANDOM MOTION WITH GAMMA-DISTRIBUTED ALTERNATING VELOCITIES IN
BIOLOGICAL MODELING
Antonio Di Crescenzo and Barbara Martinucci
Motivated by applications in mathematical biology concerning randomly
alternating motion of micro-organisms, we analyze a generalized
integrated
telegraph process. The random times between consecutive velocity
reversals are
gamma-distributed, and perform an alternating renewal process. We
obtain the
probability law and the mean of the process.
http://arxiv.org/abs/0803.1067
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6771. PLANE RECURSIVE TREES, STIRLING PERMUTATIONS AND AN URN MODEL
Svante Janson
We exploit a bijection between plane recursive trees and Stirling
permutations; this yields the equivalence of some results previously
proven
separately by different methods for the two types of objects as well
as some
new results. We also prove results on the joint distribution of the
numbers of
ascents, descents and plateaux in a random Stirling permutation. The
proof uses
an interesting generalized Polya urn
http://arxiv.org/abs/0803.1129
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6772. LONG TIME BEHAVIOUR OF A STOCHASTIC-LAGRANGIAN PARTICLE SYSTEM
FOR THE NAVIER-STOKES EQUATIONS
Gautam Iyer and Jonathan Mattingly
This paper is based on a formulation of the Navier-Stokes equations
developed
in \texttt{arxiv:math.PR/0511067} (to appear in Commun. Pure Appl.
Math), where
the velocity field of a viscous incompressible fluid is written as the
expected
value of a stochastic process. In this paper, we take $N$ copies of
the above
process (each based on independent Wiener processes), and replace the
expected
value with $\frac{1}{N}$ times the sum over these $N$ copies.
We prove that in two dimensions, this system has (time) global
solutions with
$\holderspace{1}{\alpha}$ initial data. Further, we show that as $N
\to \infty$
the system converges to the solution of Navier-Stokes equations on any
finite
interval $[0,T]$. However for fixed $N$, we prove that this system
retains
roughly $O(\frac{1}{N})$ times it's original energy $t \to \infty$.
For general
flows, we only provide a lower bound to this effect. In the special
case of
shear flows, we compute the behaviour as $t \to \infty$ explicitly.
http://arxiv.org/abs/0803.1222
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6773. PARAMETER COLLAPSE DUE TO THE ZEROS IN THE INVERSE CONDITION
R. Spjut
Helton, Lasserre, and Putinar (2008, Ann. Probability; arXiv:math/
0702314)
expose the relationship between three properties of a measure: the
conditional
triangularity property of the associated orthogonal polynomials, the
zeros in
the inverse condition of the truncated moment matrix, and conditional
independence. The purpose of this article is to provide examples of
parameter
collapse to product structure given that the zeros in the inverse
condition
holds up to some degree d. Specifically, start with a parameterized
family of
probability density functions; require that the zeros in the inverse
condition
up to degree d holds; and validate that imposing this restriction on the
parameterized family results in a measure with product structure, or
at least
that conditional independence holds. Algorithms related to parameter
collapse
are supplied, including the computation of the zeros in the inverse
condition
up to degree d.
http://arxiv.org/abs/0803.1225
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6774. ON SOME TRANSFORMATIONS OF BILATERAL BIRTH-AND-DEATH PROCESSES
WITH APPLICATIONS
Antonio Di Crescenzo
A method yielding simple relationships among bilateral birth-and-death
processes is outlined. This allows one to relate birth and death rates
of two
processes in such a way that their transition probabilities, first-
passage-time
densities and ultimate crossing probabilities are mutually related by
some
product-form expressions.
http://arxiv.org/abs/0803.1413
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6775. ESTIMATION OF WIENER--ITO INTEGRALS AND POLYNOMIALS OF
INDEPENDENT GAUSSIAN RANDOM VARIABLES
Peter Major
In this paper I prove good estimates on the moments and tail
distribution of
$k$-fold Wiener--It\^o integrals and also present their natural
counterpart for
polynomials of independent Gaussian random variables. The proof is
based on the
so-called diagram formula for Wiener--It\^o integrals which yields a
good
representation for their products as a sum of such integrals. I intend
to show
in a subsequent paper that this method also yields good estimates for
degenerate $U$-statistics. The main result of this paper is a
generalization of
the estimates of Hanson and Wright about bilinear forms of independent
standard
normal random variables. On the other hand, it is a weaker estimate
than the
main result of a paper of Lata{\l}a [6]. But that paper contains an
error, and
it is not clear whether its result is true. This question is also
discussed
here.
http://arxiv.org/abs/0803.1453
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6776. LYAPUNOV EXPONENTS FOR THE ONE-DIMENSIONAL PARABOLIC ANDERSON
MODEL WITH DRIFT
Alexander Drewitz
We consider the solution $u$ to the one-dimensional parabolic Anderson
model
with homogeneous initial condition $u(0, \cdot) \equiv 1$, arbitrary
drift and
a time-independent potential bounded from above. Under ergodicity and
independence conditions we derive representations for both the quenched
Lyapunov exponent and, more importantly, the $p$-th annealed Lyapunov
exponents
for {\it all} $p \in (0, \infty).$ These results enable us to prove the
heuristically plausible fact that the $p$-th annealed Lyapunov exponent
converges to the quenched Lyapunov exponent as $p \downarrow 0.$
Furthermore,
we show that $u$ is $p$-intermittent for $p$ large enough. As a
byproduct, we
compute the optimal quenched speed of the random walk appearing in the
Feynman-Kac representation of $u$ under the corresponding Gibbs
measure. In
this context, depending on the negativity of the potential, a phase
transition
from zero speed to positive speed appears.
http://arxiv.org/abs/0803.1480
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6777. DIFFRACTION OF STOCHASTIC POINT SETS: EXACTLY SOLVABLE EXAMPLES
Michael Baake (Bielefeld) and Matthias Birkner (Berlin) and Robert
V. Moody (Victoria)
Stochastic point sets are considered that display a diffraction
spectrum of
mixed type, with special emphasis on explicitly computable cases
together with
a unified approach of reasonable generality. Several pairs of
autocorrelation
and diffraction measures are discussed which show a duality structure
that may
be viewed as analogues of the Poisson summation formula for lattice
Dirac
combs.
http://arxiv.org/abs/0803.1266
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6778. REVERSIBILITY OF INTERACTING FLEMING-VIOT PROCESSES WITH
MUTATION, SELECTION, AND RECOMBINATION
Shui Feng and Byron Schmuland and Jean Vaillancourt and and Xiaowen
Zhou
Reversibility of the Fleming-Viot process with mutation, selection, and
recombination is well understood. In this paper, we study the
reversibility of
a system of Fleming-Viot processes that live on a countable number of
colonies
interacting with each other through migrations between the colonies.
It is
shown that reversibility fails when both migration and mutation are
non-trivial.
http://arxiv.org/abs/0803.1492
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6779. ON SOME GENERALIZED REINFORCED RANDOM WALKS ON INTEGERS
Olivier Raimond (LM-Orsay) and Bruno Schapira (LM-Orsay)
We consider Reinforced Random Walks where transition probabilities are a
function of the proportion of times the walk has traversed an edge. We
give
conditions for recurrence or transience. A phase transition is observed,
similar to Pemantle \cite{Pem000} on trees.
http://arxiv.org/abs/0803.1590
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6780. EQUALITY OF PRESSURES FOR DIFFEOMORPHISMS PRESERVING HYPERBOLIC
MEASURES
Katrin Gelfert
For a diffeomorphism which preserves a hyperbolic measure the potential
$\phi^u=-\log|{\rm Jac} df|_{E^u}|$ is studied. Various types of
pressure of
$\phi^u$ are introduced. It is shown that these pressures satisfy a
corresponding variational principle.
http://arxiv.org/abs/0803.1525
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6781. A NOTE ON MULTI-TYPE COOKIE RANDOM WALK ON INTEGERS
Bruno Schapira (LM-Orsay)
We consider a random walk on integers where at the first visits to a
site the
walker gets a positive drift, but where after a certain number of
visits the
walker gets a negative drift. We prove that the walker is almost surely
transient to the left with positive speed. This is a variant of a
model studied
by Zerner, Kosygina and Zerner, and Basdevant and Singh.
http://arxiv.org/abs/0803.1664
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6782. COPOLYMERS AT SELECTIVE INTERFACES: NEW BOUNDS ON THE PHASE
DIAGRAM
T. Bodineau and G. Giacomin and H. Lacoin and F. Toninelli
We investigate the phase diagram of disordered copolymers at the
interface
between two selective solvents, and in particular its weak-coupling
behavior,
encoded in the slope $m_c$ of the critical line at the origin. In
mathematical
terms, the partition function of such a model does not depend on all the
details of the Markov chain that models the polymer, but only on the
time
elapsed between successive returns to zero and on whether the walk is
in the
upper or lower half plane between such returns. This observation leads
to a
natural generalization of the model, in terms of arbitrary laws of
return
times: the most interesting case being the one of return times with
power law
tails (with exponent $1+\ga$, $\ga=1/2$ in the case of the symmetric
random
walk). The main results we present here are:
1/ The improvement of the known result $1/(1+\alpha)\le m_c\le 1$,
as soon as
$\ga >1$ for what concerns the upper bound, and down to $\ga\approx
0.65$ for
the lower bound.
2/ A proof of the fact that the critical curve lies strictly below
the
critical curve of the annealed model for every non-zero value of the
coupling
parameter.
We also provide an argument that rigorously shows the strong
dependence of
the phase diagram on the details of the return probability (and not
only on the
tail behavior). Lower bounds are obtained by exhibiting a new
localization
strategy, while upper bounds are based on estimates of non-integer
moments of
the partition function.
http://arxiv.org/abs/0803.1766
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6783. BSDES WITH TWO RCLL REFLECTING OBSTACLES DRIVEN BY A BROWNIAN
MOTION AND POISSON MEASURE AND RELATED MIXED ZERO-SUM GAMES
S.Hamad\'ene and H.Wang
In this paper we study Backward Stochastic Differential Equations with
two
reflecting right continuous with left limits obstacles (or barriers)
when the
noise is given by Brownian motion and a Poisson random measure mutually
independent. The jumps of the obstacle processes could be either
predictable or
inaccessible. We show existence and uniqueness of the solution when the
barriers are completely separated and the generator uniformly
Lipschitz. We do
not assume the existence of a difference of supermartingales between the
obstacles. As an application, we show that the related mixed zero-sum
differential-integral game problem has a value.
http://arxiv.org/abs/0803.1815
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6784. BALANCE, GROWTH AND DIVERSITY OF FINANCIAL MARKETS
Constantinos Kardaras
A financial market comprising of a certain number of distinct
companies is
considered, and the following statement is proved: either a specific
agent will
surely beat the whole market unconditionally in the long run, or (and
this "or"
is not exclusive) all the capital of the market will accumulate in one
company.
Thus, absence of any "free unbounded lunches relative to the total
capital"
opportunities lead to the most dramatic failure of diversity in the
market: one
company takes over all other until the end of time. In order to prove
this, we
introduce the notion of perfectly balanced markets, which is an
equilibrium
state in which the relative capitalization of each company is a
martingale
under the physical probability. Then, the weaker notion of balanced
markets is
discussed where the martingale property of the relative
capitalizations holds
only approximately, we show how these concepts relate to growth-
optimality and
efficiency of the market, as well as how we can infer a shadow
interest rate
that is implied in the economy in the absence of a bank.
http://arxiv.org/abs/0803.1858
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6785. THE NUMERAIRE PORTFOLIO IN SEMIMARTINGALE FINANCIAL MODELS
Ioannis Karatzas and Constantinos Kardaras
We study the existence of the numeraire portfolio under predictable
convex
constraints in a general semimartingale model of a financial market. The
numeraire portfolio generates a wealth process, with respect to which
the
relative wealth processes of all other portfolios are supermartingales.
Necessary and sufficient conditions for the existence of the numeraire
portfolio are obtained in terms of the triplet of predictable
characteristics
of the asset price process. This characterization is then used to obtain
further necessary and sufficient conditions, in terms of a no-free-
lunch-type
notion. In particular, the full strength of the "No Free Lunch with
Vanishing
Risk" (NFLVR) is not needed, only the weaker "No Unbounded Profit with
Bounded
Risk" (NUPBR) condition that involves the boundedness in probability
of the
terminal values of wealth processes. We show that this notion is the
minimal
a-priori assumption required in order to proceed with utility
optimization. The
fact that it is expressed entirely in terms of predictable
characteristics
makes it easy to check, something that the stronger NFLVR condition
lacks.
http://arxiv.org/abs/0803.1877
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6786. MONOTONICITY FOR EXCITED RANDOM WALK IN HIGH DIMENSIONS
Remco van der Hofstad and Mark Holmes
We prove that the drift $\theta(d,\beta)$ for excited random walk in
dimension $d$ is monotone in the excitement parameter $\beta \in[0,
1]$, when
$d\ge 9$.
http://arxiv.org/abs/0803.1881
---------------------------------------------------------------
6787. ON FINANCIAL MARKETS WHERE ONLY BUY-AND-HOLD TRADING IS POSSIBLE
Constantinos Kardaras and Eckhard Platen
A financial market model where agents can only trade using realistic
buy-and-hold trategies is considered. Minimal assumptions are made on
the
nature of the asset-price process - in particular, the semimartingale
property
is not assumed. Via a natural assumption of limited opportunities for
unlimited
resulting wealth from trading, coined the No-Unbounded-Profit-with-
Bounded-Risk
(NUPBR) condition, we establish that asset-prices have be
semimartingales, as
well as a weakened version of the Fundamental Theorem of Asset Pricing
that
involves supermartingale deflators rather than equivalent martingale
measures.
Further, the utility maximization problem is considered and it is
shown that
using only buy-and-hold strategies, optimal utilities and wealth
processes
resulting from continuous trading can be approximated arbitrarily well.
http://arxiv.org/abs/0803.1890
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6788. CONSTANTS OF CONCENTRATION FOR A SIMPLE RECURRENT RANDOM WALK ON
RANDOM ENVIRONMENT
Pierre Andreoletti (MAPMO)
We precise the asymptotic of the limsup of the size of the
neighborhood of
concentration of Sinai's walk. Also we get the almost sure limits of
the number
of points visited more than a fixed proportion of a given amount of
time.
http://arxiv.org/abs/0803.2006
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6789. STRONG LAW OF LARGE NUMBERS WITH CONCAVE MOMENTS
Anders Karlsson and Nicolas Monod
In this note not intended for publication, it is observed that a
wellnigh
trivial application of the ergodic theorem of Karlsson-Ledrappier
yields a
strong LLN for arbitrary concave moments.
http://arxiv.org/abs/0803.1856
---------------------------------------------------------------
6790. DIFFUSION AT THE RANDOM MATRIX HARD EDGE
Jose A. Ramirez and Brian Rider
We show that the limiting minimal eigenvalue distributions for a natural
generalization of Gaussian sample-covariance structures (the "beta
ensembles")
are described in by the spectrum of a random diffusion generator. By a
Riccati
transformation, we obtain a second diffusion description of the limiting
eigenvalues in terms of hitting laws. This picture pertains to the so-
called
hard edge of random matrix theory and sits in complement to the recent
work of
the authors and B. Virag on the general beta random matrix soft edge.
In fact,
the diffusion descriptions found on both sides are used here to prove
there
exists a transition between the soft and hard edge laws at all values
of beta.
http://arxiv.org/abs/0803.2043
---------------------------------------------------------------
6791. OPTIMAL TWO-VALUE ZERO-MEAN DISINTEGRATION OF ZERO-MEAN RANDOM
VARIABLES
Iosif Pinelis
For any continuous zero-mean random variable (r.v.) X, a reciprocating
function r is constructed, based only on the distribution of X, such
that the
conditional distribution of X given the (at-most-)two-point set
{X,r(X)} is the
zero-mean distribution on this set; in fact, a more general construction
without the continuity assumption is given in this paper, as well as a
large
variety of other related results, including characterizations of the
reciprocating function and modeling distribution asymmetry patterns. The
mentioned disintegration of zero-mean r.v.'s implies, in particular,
that an
arbitrary zero-mean distribution is represented as the mixture of two-
point
zero-mean distributions; moreover, this mixture representation is most
symmetric in a variety of senses. Somewhat similar representations --
of any
probability distribution as the mixture of two-point distributions
with the
same skewness coefficient (but possibly with different means) -- go
back to
Kolmogorov; very recently, Aizenman et al. further developed such
representations and applied them to (anti-)concentration inequalities
for
functions of independent random variables and to spectral localization
for
random Schroedinger operators. One kind of application given in the
present
paper is to construct certain statistical tests for asymmetry patterns
and for
location without symmetry conditions. Exact inequalities implying
conservative
properties of such tests are presented. These developments extend
results
established earlier by Efron, Eaton, and Pinelis under a symmetry
condition.
http://arxiv.org/abs/0803.2068
---------------------------------------------------------------
6792. NO-FREE-LUNCH EQUIVALENCES FOR EXPONENTIAL LEVY MODELS
Constantinos Kardaras
We provide equivalence of numerous no-free-lunch type conditions for
financial markets where the asset prices are modeled as exponential Levy
processes, under possible convex constraints in the use of investment
strategies. The general message is the following: if any kind of free
lunch
exists in these models it has to be of the most egregious type,
generating an
increasing ealth. Furthermore, we connect the previous to the
existence of the
numeraire portfolio, both for its particular expositional clarity in
exponential Levy models and as a first step in obtaining analogues of
the
no-free-lunch equivalences in general semimartingale models.
http://arxiv.org/abs/0803.2169
---------------------------------------------------------------
6793. ON AGENTS' AGREEMENT AND PARTIAL-EQUILIBRIUM PRICING IN
INCOMPLETE MARKETS
Michail Anthropelos and Gordan Zitkovic
We consider two risk-averse financial agents who negotiate the price
of an
illiquid indivisible contingent claim in an incomplete semimartingale
market
environment. Under the assumption that the agents are exponential
utility
maximizers with non-traded random endowments, we provide necessary and
sufficient conditions for negotiation to be successful, i.e., for the
trade to
occur. We also study the asymptotic case where the size of the claim
is small
compared to the random endowments and we give a full characterization
in this
case. Finally, we study a partial-equilibrium problem for a bundle of
divisible
claims and establish existence and uniqueness. A number of technical
results on
conditional indifference prices are provided.
http://arxiv.org/abs/0803.2198
---------------------------------------------------------------
6794. DUALITY OF CHORDAL SLE, II
Dapeng Zhan
We improve the geometric properties of SLE$(\kappa;\vec{\rho})$
processes
derived in an earlier paper, which are then used to obtain more
results about
the duality of SLE. We find that for $\kappa\in (4,8)$, the boundary
of a
standard chordal SLE$(\kappa)$ hull stopped on swallowing a fixed
$x\in\R\sem\{0\}$ is the image of some SLE$(16/\kappa;\vec{\rho})$ trace
started from a random point. Using this fact together with a similar
proposition in the case that $\kappa\ge 8$, we obtain a description of
the
boundary of a standard chordal SLE$(\kappa)$ hull for $\kappa>4$, at a
finite
stopping time. Finally, we prove that for $\kappa>4$, in many cases,
the limit
of a chordal or strip SLE$(\kappa;\vec{\rho})$ trace exists.
http://arxiv.org/abs/0803.2223
---------------------------------------------------------------
6795. A DECOMPOSITION OF THE BIFRACTIONAL BROWNIAN MOTION AND SOME
APPLICATIONS
Pedro Lei and David Nualart
In this paper we show a decomposition of the bifractional Brownian
motion
with parameters H,K into the sum of a fractional Brownian motion with
Hurst
parameter HK plus a stochastic process with absolutely continuous
trajectories.
Some applications of this decomposition are discussed.
http://arxiv.org/abs/0803.2227
---------------------------------------------------------------
6796. RANDOM SOLUTIONS OF RANDOM PROBLEMS...ARE NOT JUST RANDOM
Dimitris Achlioptas and Amin Coja-Oghlan
Let I(n,m) denote a uniformly random instance of some constraint
satisfaction
problem CSP with n variables and m constraints. Assume that the
density r=m/n
is small enough so that with high probability I(n,m) has a solution, and
consider the experiment of first choosing an instance I=I(n,m) at
random, and
then sampling a random solution sigma of I (if one exists). For many
CSPs
(e.g., k-SAT, k-NAE, or k-coloring), this experiment appears difficult
both to
implement and to analyze; in fact, for a large range of r, no efficient
algorithm is known to even compute a single solution of I. In the
present paper
we show that for many CSPs the above experiment is essentially
equivalent to
first choosing a random assignment sigma to the n variables, and then
drawing a
random instance satisfied by sigma uniformly. In general, this second
experiment is very easy to implement and amenable to a rigorous
analysis. In
fact, using this equivalence, we can analyze the solution space of
random CSPs.
Thus, we can achieve the long-standing goal of establishing rigorously a
picture put forward by statistical physicists on the basis of
sophisticated but
non-rigorous techniques such as the cavity and the replica method.
This picture
is suggestive as to why random CSP instances seem difficult to deal with
algorithmically. Furthermore, we show that the second experiment gives
rise to
one-way functions, if one assumes that random instances of CSP are
hard for
some range of densities.
http://arxiv.org/abs/0803.2122
---------------------------------------------------------------
6797. PARTIAL DIFFERENTIAL EQUATIONS DRIVEN BY ROUGH PATHS
Michael Caruana and Peter Friz
We study a class of linear first and second order partial differential
equations driven by weak geometric $p$-rough paths, and prove the
existence of
a unique solution for these equations. This solution depends
continuously on
the driving rough path. This allows a robust approach to stochastic
partial
differential equations. In particular, we may replace Brownian motion
by more
general Gaussian and Markovian noise. Support theorems and large
deviation
statements all became easy corollaries of the corresponding statements
of the
driving process. In the case of first order equations with Gaussian
noise, we
discuss the existence of a density with respect to the Lebesgue
measure for the
solution.
http://arxiv.org/abs/0803.2178
---------------------------------------------------------------
6798. ON PERPETUAL AMERICAN PUT VALUATION AND FIRST-PASSAGE IN A
REGIME-SWITCHING MODEL WITH JUMPS
Z. Jiang and M.R. Pistorius
In this paper we consider the problem of pricing a perpetual American
put
option in an exponential regime-switching L\'{e}vy model. For the case
of the
(dense) class of phase-type jumps and finitely many regimes we derive an
explicit expression for the value function. The solution of the
corresponding
first passage problem under a state-dependent level rests on a path
transformation and a new matrix Wiener-Hopf factorization result for
this class
of processes.
http://arxiv.org/abs/0803.2302
---------------------------------------------------------------
6799. ABSOLUTE CONTINUITY AND CONVERGENCE IN VARIATION FOR
DISTRIBUTIONS OF A FUNCTIONALS OF POISSON POINT MEASURE
Alexey M.Kulik
General sufficient conditions are given for absolute continuity and
convergence in variation of distributions of a functionals on a
probability
space, generated by a Poisson point measure. The phase space of the
Poisson
point measure is supposed to be of the form (0,\infty)\times U, and its
intensity measure to be equal dt\Pi(du). We introduce the family of time
stretching transformations of the configurations of the point measure.
The
sufficient conditions for absolute continuity and convergence in
variation are
given in the terms of the time stretching transformations and the
relative
differential operators. These conditions are applied to solutions of
SDE's
driven by Poisson point measures, including an SDE's with non-constant
jump
rate.
http://arxiv.org/abs/0803.2389
---------------------------------------------------------------
6800. ON A SZEGO TYPE LIMIT THEOREM, THE HOLDER-YOUNG-BRASCAMP-LIEB
INEQUALITY, AND THE ASYMPTOTIC THEORY OF INTEGRALS AND QUADRATIC FORMS
OF
STATIONARY FIELDS
Florin Avram (LMA-PAU) and Nikolai Leonenko and Ludmila Sakhno
Many statistical applications require establishing central limit
theorems for
sums, integrals, or for quadratic forms of functions of a stationary
process. A
particularly important case is that of Appell polynomials, since the
Appell
expansion rank" determines typically the type of central limit theorem
satisfied by these functionals. We review and extend here to
multidimensional
indices a functional analysis approach to this problem proposed by
Avram and
Brown (1989), based on the method of cumulants and on integrability
assumptions
in the spectral domain; several applications are presented as well.
http://arxiv.org/abs/0803.2441
---------------------------------------------------------------
6801. STABILITY OF A PROCESSOR SHARING QUEUE WITH VARYING THROUGHPUT
Pascal Moyal
In this paper, we present a stability criterion for Processor Sharing
queues,
in which the throughput may depend on the number of customers in the
system (in
such cases such as interferences between the users). Such a system is
represented by a point measure-valued stochastic recursion keeping
track of the
remaining processing times of the customers.
http://arxiv.org/abs/0803.2459
---------------------------------------------------------------
6802. ON THE LEAST SQUARES ESTIMATOR IN A NEARLY UNSTABLE SEQUENCE OF
STATIONARY SPATIAL AR MODELS
S\'andor Baran and Gyula Pap
A nearly unstable sequence of stationary spatial autoregressive
processes is
investigated, when the sum of the absolute values of the autoregressive
coefficients tends to one. It is shown that after an appropriate
norming the
least squares estimator for these coefficients has a normal limit
distribution.
If none of the parameters equals zero than the typical rate of
convergence is
n.
http://arxiv.org/abs/0803.2486
---------------------------------------------------------------
6803. A RUELLE OPERATOR FOR CONTINUOUS TIME MARKOV CHAINS
Alexandre Baraviera and Ruy Exel and Artur O. Lopes
We consider a generalization of the Ruelle theorem for the case of
continuous
time problems. We present a result which we believe is important for
future use
in problems in Mathematical Physics related to $C^*$-Algebras We
consider a
finite state set $S$ and a stationary continuous time Markov Chain $X_t
$,
$t\geq 0$, taking values on $S$. We denote by $\Omega$ the set of
paths $w$
taking values on $S$ (the elements $w$ are locally constant with left
and right
limits and are also right continuous on $t$). We consider an
infinitesimal
generator $L$ and a stationary vector $p_0$. We denote by $P$ the
associated
probability on ($\Omega, {\cal B}$). All functions $f$ we consider
bellow are
in the set ${\cal L}^\infty (P)$. From the probability $P$ we define a
Ruelle
operator ${\cal L}^t, t\geq 0$, acting on functions $f:\Omega \to
\mathbb{R}$
of ${\cal L}^\infty (P)$. Given $V:\Omega \to \mathbb{R}$, such that is
constant in sets of the form $\{X_0=c\}$, we define a modified Ruelle
operator
$\tilde{{\cal L}}_V^t, t\geq 0$. We are able to show the existence of an
eigenfunction $u$ and an eigen-probability $\nu_V$ on $\Omega$
associated to
$\tilde{{\cal L}}^t_V, t\geq 0$. We also show the following property
for the
probability $\nu_V$: for any integrable $g\in {\cal L}^\infty (P)$ and
any real
and positive $t$ $$ \int e^{-\int_0^t (V \circ \Theta_s)(.) ds}
[ (\tilde{{\cal
L}}^t_V (g)) \circ \theta_t ] d \nu_V = \int g d \nu_V$$ This equation
generalize, for the continuous time Markov Chain, a similar one for
discrete
time systems (and which is quite important for understanding the KMS
states of
certain $C^*$-algebras).
http://arxiv.org/abs/0803.2501
---------------------------------------------------------------
6804. A CIRCLE OF INTERACTING SERVERS; SPONTANEOUS COLLECTIVE BEHAVIOR
IN CASE OF LARGE FLUCTUATIONS
E.A. Pechersky and N.D. Vvedenskaya
We consider large fluctuations, namely overload of servers, in a
network with
dynamic routing of messages. The servers form a circle. The number of
input
flows is equal to the number of servers, the messages of any flow are
distributed between two neighboring servers, upon its arrival a
message is
directed to the least loaded of these servers. Under the condition
that at
least two servers are overloaded the number of overloaded servers in
such
network depends on the rate of input flows. In particular there exists
critical
level of input rate that in case of higher rate most probable that all
servers
are overloaded.
http://arxiv.org/abs/0803.2576
---------------------------------------------------------------
6805. MARKOV CHAINS APPROXIMATIONS OF JUMP-DIFFUSION QUANTUM
TRAJECTORIES
Clement Pellegrini (ICJ)
"Quantum trajectories" are solutions of stochastic differential
equations
also called Belavkin or Stochastic Schr\"odinger Equations. They
describe
random phenomena in quantum measurement theory. Two types of such
equations are
usually considered, one is driven by a one-dimensional Brownian motion
and the
other is driven by a counting process. In this article, we present a
way to
obtain more advanced models which use jump-diffusion stochastic
differential
equations. Such models come from solutions of martingale problems for
infinitesimal generators. These generators are obtained from the limit
of
generators of classical Markov chains which describe discrete models
of quantum
trajectories. Furthermore, stochastic models of jump-diffusion
equations are
physically justified by proving that their solutions can be obtained
as the
limit of the discrete trajectories.
http://arxiv.org/abs/0803.2593
---------------------------------------------------------------
6806. POISSON AND DIFFUSION APPROXIMATION OF STOCHASTIC SCHRODINGER
EQUATIONS WITH CONTROL
Clement Pellegrini (ICJ)
"Quantum trajectories" are solutions of stochastic differential
equations of
non-usual type. Such equations are called ``Belavkin'' or ``Stochastic
Schr\"odinger Equations'' and describe random phenomena in continuous
measurement theory of Open Quantum System. Many recent investigations
deal with
the control theory in such model. In this article, stochastic models are
mathematically and physically justified as limit of concrete discrete
procedures called ``Quantum Repeated Measurements''. In particular,
this gives
a rigorous justification of the Poisson and diffusion approximation in
quantum
measurement theory with control. Furthermore we investigate some
examples using
control in quantum mechanics.
http://arxiv.org/abs/0803.2643
---------------------------------------------------------------
6807. A NEW CENTRAL LIMIT THEOREM UNDER SUBLINEAR EXPECTATIONS
Shige Peng
We describe a new framework of a sublinear expectation space and the
related
notions and results of distributions, independence. A new notion of
G-distributions is introduced which generalizes our G-normal-
distribution in
the sense that mean-uncertainty can be also described. W present our
new result
of central limit theorem under sublinear expectation. This theorem can
be also
regarded as a generalization of the law of large number in the case of
mean-uncertainty.
http://arxiv.org/abs/0803.2656
---------------------------------------------------------------
6808. FIELD THEORY CONJECTURE FOR LOOP-ERASED RANDOM WALKS
Andrei A. Fedorenko and Pierre Le Doussal and Kay Joerg Wiese
We give evidence that the functional renormalization group (FRG),
developed
to study disordered systems, may provide a field theoretic description
for the
loop-erased random walk (LERW), allowing to compute its fractal
dimension in a
systematic expansion in epsilon=4-d. Up to two loop, the FRG agrees with
rigorous bounds, correctly reproduces the leading logarithmic
corrections at
the upper critical dimension d=4, and compares well with numerical
studies. We
obtain the universal subleading logarithmic correction in d=4, which
can be
used as a further test of the conjecture.
http://arxiv.org/abs/0803.2357
---------------------------------------------------------------
6809. DISCRETE STOCHASTIC PROCESSES, REPLICATOR AND FOKKER-PLANCK
EQUATIONS OF COEVOLUTIONARY DYNAMICS IN FINITE AND INFINITE POPULATIONS
Jens Christian Claussen
Finite-size fluctuations in coevolutionary dynamics arise in models of
biological as well as of social and economic systems. This brief
tutorial
review surveys a systematic approach starting from a stochastic process
discrete both in time and state. The limit $N\to \infty$ of an infinite
population can be considered explicitly, generally leading to a
replicator-type
equation in zero order, and to a Fokker-Planck-type equation in first
order in
$1/\sqrt{N}$. Consequences and relations to some previous approaches are
outlined.
http://arxiv.org/abs/0803.2443
---------------------------------------------------------------
6810. TYPICAL DISPERSION AND GENERALIZED LYAPUNOV EXPONENTS
Steven Finch and Zai-Qiao Bai and Pascal Sebah
Let f(n) denote the number of odd entries in the nth row of Pascal's
binomial
triangle. We study "average dispersion" and "typical dispersion" of
f(n) -- the
latter involves computing a generalized Lyapunov exponent -- and then
turn to
numerical analysis of higher dimensional examples.
http://arxiv.org/abs/0803.2611
---------------------------------------------------------------
6811. POTTS MODELS IN THE CONTINUUM. UNIQUENESS AND EXPONENTIAL DECAY
IN THE RESTRICTED ENSEMBLES
A. De Masi and I. Merola and E. Presutti and Y. Vignaud
In this paper we study a continuum version of the Potts model.
Particles are
points in R^d, with a spin which may take S possible values, S being
at least
3. Particles with different spins repel each other via a Kac pair
potential. In
mean field, for any inverse temperature there is a value of the chemical
potential at which S+1 distinct phases coexist. For each mean field
pure phase,
we introduce a restricted ensemble which is defined so that the
empirical
particles densities are close to the mean field values. Then, in the
spirit of
the Dobrushin Shlosman theory, we get uniqueness and exponential decay
of
correlations when the range of the interaction is large enough. In a
second
paper, we will use such a result to implement the Pirogov-Sinai scheme
proving
coexistence of S+1 extremal DLR measures.
http://arxiv.org/abs/0803.2767
---------------------------------------------------------------
6812. ON MONGE-KANTOROVICH PROBLEM IN THE PLANE
Yinfang Shen and Weian Zheng
We transfer the celebrating Monge-Kontorovich problem in a bounded
domain of
Euclidean plane into a Dirichlet boundary problem associated to a
quasi-linear
elliptic equation with $0-$order term missing in its diffusion
coefficients:
\begin{eqnarray*} A(x, F'_x)F''_{xx}+B(y, F'_y)F''_{yy}&=&C(x, y,
F'_x, F'_y)
\end{eqnarray*} where $A(.,.)>0, B(.,.)>0$ and $C$ are functions based
on the
initial distributions, $F$ is an unknown probability distribution
function and
therefore closed the former problem.
http://arxiv.org/abs/0803.2830
---------------------------------------------------------------
6813. SELF-REPELLING RANDOM WALK WITH DIRECTED EDGES ON Z
Balint Toth and Balint Veto
We consider a variant of self-repelling random walk on the integer
lattice Z
where the self-repellence is defined in terms of the local time on
oriented
edges. The long-time asymptotic scaling of this walk is surprisingly
different
from the asymptotics of the similar process with self-repellence
defined in
terms of local time on unoriented edges. We prove limit theorems for
the local
time process and for the position of the random walker. The main
ingredient is
a Ray-Knight-type of approach. At the end of the paper, we also
present some
computer simulations which show the strange scaling behaviour of the
walk
considered.
http://arxiv.org/abs/0803.2848
---------------------------------------------------------------
6814. PRODUCT-FORM STATIONARY DISTRIBUTIONS FOR DEFICIENCY ZERO
CHEMICAL REACTION NETWORKS
David F. Anderson and Gheorghe Craciun and Thomas G. Kurtz
We consider both deterministically and stochastically modeled chemical
reaction systems and prove that a product-form stationary distribution
exists
for each closed, irreducible subset of the state space of a
stochastically
modeled system if the corresponding deterministically modeled system
admits a
complex balanced equilibrium. Feinberg's deficiency zero theorem then
implies
that such a distribution exists so long as the corresponding chemical
network
is weakly reversible and has a deficiency of zero. We also demonstrate
that the
main parameter of the stationary distribution for the stochastically
modeled
system is a complex balanced equilibrium value for the corresponding
deterministically modeled system.
http://arxiv.org/abs/0803.3042
---------------------------------------------------------------
6815. REPEATED QUANTUM INTERACTIONS QUANTUM LANGEVIN EQUATION AND THE
LOW DENSITY LIMIT
Ameur Dhahri (CEREMADE)
We consider a repeated quantum interaction model describing a small
system
$\Hh_S$ in interaction with each one of the identical copies of the
chain
$\bigotimes_{\N^*}\C^{n+1}$, modeling a heat bath, one after another
during the
same short time intervals $[0,h]$. We suppose that the repeated quantum
interaction Hamiltonian is split in two parts: a free part and an
interaction
part with time scale of order $h$. After giving the GNS
representation, we
establish the relation between the time scale $h$ and the classical
low density
limit. We introduce a chemical potential $\mu$ related to the time $h$
as
follows: $h^2=e^{\beta\mu}$. We further prove that the solution of the
associated discrete evolution equation converges strongly, when $h$
tends to 0,
to the unitary solution of a quantum Langevin equation directed by
Poisson
processes.
http://arxiv.org/abs/0803.3059
---------------------------------------------------------------
6816. A LINDBLAD MODEL FOR A SPIN CHAIN COUPLED TO HEAT BATHS
Ameur Dhahri (ICJ and Ceremade)
We study a XY model which consists of a spin chain coupled to heat
baths. We
give a repeated quantum interaction Hamiltonian describing this model.
We
compute the explicit form of the associated Lindblad generator in the
case of
the spin chain coupled to one, two and several heat baths. We further
study the
properties of quantum master equation such as approach to equilibrium,
local
equilibrium states, entropy production and quantum detailed balance
condition.
http://arxiv.org/abs/0803.3060
---------------------------------------------------------------
6817. RECORDS IN A CHANGING WORLD
Joachim Krug
In the context of this paper, a record is an entry in a sequence of
random
variables (RV's) that is larger or smaller than all previous entries.
After a
brief review of the classic theory of records, which is largely
restricted to
sequences of independent and identically distributed (i.i.d.) RV's,
new results
for sequences of independent RV's with distributions that broaden or
sharpen
with time are presented. In particular, we show that when the width of
the
distribution grows as a power law in time $n$, the mean number of
records is
asymptotically of order $\ln n$ for distributions with a power law
tail (the
\textit{Fr\'echet class} of extremal value statistics), of order $(\ln
n)^2$
for distributions of exponential type (\textit{Gumbel class}), and of
order
$n^{1/(\nu+1)}$ for distributions of bounded support (\textit{Weibull
class}),
where the exponent $\nu$ describes the behaviour of the distribution
at the
upper (or lower) boundary. Simulations are presented which indicate
that, in
contrast to the i.i.d. case, the sequence of record breaking events is
correlated in such a way that the variance of the number of records is
asymptotically smaller than the mean.
http://arxiv.org/abs/cond-mat/0702136
---------------------------------------------------------------
6818. LOCAL SEMICIRCLE LAW AND COMPLETE DELOCALIZATION FOR WIGNER
RANDOM MATRICES
Laszlo Erdos and Benjamin Schlein and Horng-Tzer Yau
We consider $N\times N$ Hermitian random matrices with i.i.d. entries.
The
matrix is normalized so that the average spacing between consecutive
eigenvalues is of order 1/N. Under suitable assumptions on the
distribution of
the single matrix element, we prove that, away from the spectral
edges, the
density of eigenvalues concentrates around the Wigner semicircle law
on energy
scales $\eta \gg N^{-1} (\log N)^8$. Up to the logarithmic factor,
this is the
smallest energy scale for which the semicircle law may be valid. We
also prove
that for all eigenvalues away from the spectral edges, the $\ell^\infty
$-norm
of the corresponding eigenvectors is of order $O(N^{-1/2})$, modulo
logarithmic
corrections. The upper bound $O(N^{-1/2})$ implies that every
eigenvector is
completely de-localized, i.e., the maximum size of the components of the
eigenvector is of the same order as their average size. In the
Appendix, we
include a lemma by J. Bourgain which removes one of our assumptions on
the
distribution of the matrix elements.
http://arxiv.org/abs/0803.0542
---------------------------------------------------------------
6819. INTERACTING PARTICLE SYSTEMS OUT OF EQUILIBRIUM
Thomas Kriecherbauer and Joachim Krug
These notes are based on lectures delivered by the authors at the
Langeoog
seminar of SFB/TR12 "Symmetries and universality in mesoscopic
systems" in
November 2007, to a mixed audience of mathematicians and theoretical
physicists. After a brief outline of the basic physical concepts of
equilibrium
and nonequilibrium states, the one-dimensional totally asymmetric simple
exclusion process (TASEP) is introduced as a paradigmatic nonequilibrium
interacting particle system. The stationary measure on the ring is
derived and
the idea of the hydrodynamic limit is sketched. We then explain in
detail a
famous rigorous result due to Johansson, which relates the TASEP current
fluctuations to the Tracy-Widom distribution of random matrix theory,
and
discuss its implications within the framework of the phenomenological
Kardar-Parisi-Zhang equation.
http://arxiv.org/abs/0803.2796
---------------------------------------------------------------
6820. MINORS IN RANDOM REGULAR GRAPHS
N. Fountoulakis and D. K\"uhn and D. Osthus
We show that there is a constant c>0 so that for any fixed r which is at
least 3 a.a.s. an r-regular graph on n vertices contains a complete
graph on c
n^{1/2} vertices as a minor. This confirms a conjecture of Markstrom.
Since any
minor of an r-regular graph on n vertices has at most rn/2 edges, our
bound is
clearly best possible up to the value of the constant c. As a
corollary, we
also obtain the likely order of magnitude of the largest complete
minor in a
random graph G(n,p) during the phase transition (i.e. when pn is close
to 1).
http://arxiv.org/abs/0803.3001
---------------------------------------------------------------
6821. DIVERSITY AND RELATIVE ARBITRAGE IN EQUITY MARKETS
Robert Fernholz and Ioannis Karatzas and Constantinos Kardaras
A financial market is called "diverse" if no single stock is ever
allowed to
dominate the entire market in terms of relative capitalization. In the
context
of the standard Ito-process model initiated by Samuelson (1965) we
formulate
this property (and the allied, successively weaker notions of "weak
diversity"
and "asymptotic weak diversity") in precise terms. We show that
diversity is
possible to achieve, but delicate. Several illustrative examples are
provided,
which demonstrate that weakly-diverse financial markets contain relative
arbitrage opportunities: it is possible to outperform (or
underperform) such
markets over sufficiently long time-horizons, and to underperform them
significantly over arbitrary time-horizons. The existence of such
relative
arbitrage does not interfere with the development of option pricing,
and has
interesting consequences for the pricing of long-term warrants and for
put-call
parity. Several open questions are suggested for further study.
http://arxiv.org/abs/0803.3093
---------------------------------------------------------------
6822. REGENERATIVE TREE GROWTH: BINARY SELF-SIMILAR CONTINUUM RANDOM
TREES AND POISSON-DIRICHLET COMPOSITIONS
Jim Pitman and Matthias Winkel
We use a natural ordered extension of the Chinese Restaurant Process
to grow
a two-parameter family of binary self-similar continuum fragmentation
trees. We
provide an explicit embedding of Ford's sequence of alpha model trees
in the
continuum tree which we identified in a previous article as a
distributional
scaling limit of Ford's trees. In general, the Markov branching trees
induced
by the two-parameter growth rule are not sampling consistent, so the
existence
of compact limiting trees cannot be deduced from previous work on the
sampling
consistent case. We develop here a new approach to establish such
limits, based
on regenerative interval partitions and the urn-model description of
sampling
from Dirichlet random distributions.
http://arxiv.org/abs/0803.3098
---------------------------------------------------------------
6823. A FEW REMARKS ON THE OPERATOR NORM OF RANDOM TOEPLITZ MATRICES
Rados{\l}aw Adamczak
We present some results concerning the almost sure behaviour of the
operator
norm or random Toeplitz matrices, including the law of large numbers
for the
norm, normalized by its expectation (in the i.i.d. case). As tools we
present
some concentration inequalities for suprema of empirical processes,
which are
refinements of recent results by Einmahl and Li.
http://arxiv.org/abs/0803.3111
---------------------------------------------------------------
6824. SYMMETRIC JUMP PROCESSES: LOCALIZATION, HEAT KERNELS, AND
CONVERGENCE
Richard F. Bass and Moritz Kassmann and and Takashi Kumagai
We consider symmetric processes of pure jump type. We prove local
estimates
on the probability of exiting balls, the H\"older continuity of harmonic
functions and of heat kernels, and convergence of a sequence of such
processes.
http://arxiv.org/abs/0803.3164
---------------------------------------------------------------
6825. ON SOME RESULTS OF CUFARO PETRONI ABOUT STUDENT T-PROCESSES
C. Berg and C. Vignat
This paper deals with Student t-processes as studied in (Cufaro
Petroni N
2007 J. Phys. A, Math. Theor. 40(10), 2227-2250). We prove and extend
some
conjectures expressed by Cufaro Petroni about the asymptotical
behavior of a
Student t-process and the expansion of its density. First, the explicit
asymptotic behavior of any real positive convolution power of a Student
t-density with any real positive degrees of freedom is given in the
multivariate case; then the integer convolution power of a Student
t-distribution with odd degrees of freedom is shown to be a convex
combination
of Student t-densities with odd degrees of freedom. At last, we show
that this
result does not extend to the case of non-integer convolution powers.
http://arxiv.org/abs/0803.3198
---------------------------------------------------------------
6826. RECURRENCE AND TRANSIENCE OF A MULTI-EXCITED RANDOM WALK ON A
REGULAR TREE
Anne-Laure Basdevant and Arvind Singh
We study a model of multi-excited random walk on a regular tree which
generalizes the models of the once excited random walk and the digging
random
walk introduced by Volkov (2003). We show the existence of a phase
transition
of the recurrence/transience property of the walk. In particular, we
prove that
the asymptotic behavior of the walk depends on the order of the
excitations,
which contrasts with the one dimensional setting studied by Zerner
(2005).
Special attention is given to the cases of the once excited, the twice
excited
and the digging random walk where explicit criterions, depending on
the initial
cookie environment, are provided to determine whether the walk is
recurrent or
transient.
http://arxiv.org/abs/0803.3284
---------------------------------------------------------------
6827. BRANCHING PROCESS APPROACH FOR 2-SAT THRESHOLDS
Elchanan Mossel (UC Berkeley) and Arnab Sen (UC Berkeley)
It is well known that, as $n$ tends to infinity, the probability of
satisfiability for a random 2-SAT formula on $n$ variables, where each
clause
occurs independently with probability $\alpha/2n$, exhibits a sharp
threshold
at $\alpha=1$. We provide a simple conceptual proof of this fact based
on
branching process arguments. We also study a generalized 2-SAT model
in which
each clause occurs independently but with probability $\alpha_i/2n$
where $i
\in \{0,1,2 \}$ is the number of positive literals in that clause. We
use
2-type branching process arguments to determine the satisfiability
threshold
for this model in terms of the maximum eigenvalue of the branching
matrix.
http://arxiv.org/abs/0803.3285
---------------------------------------------------------------
6828. A GENERALIZED FEYNMAN-KAC FORMULA FOR ONE DIMENSIONAL PROCESSES
George Lowther
Suppose that a real valued process X is given as a solution to a
stochastic
differential equation. Then, for any twice continuously differentiable
function
f, the Feynman-Kac formula gives a condition for f(t,X) to be a local
martingale.
We generalize the Feynman-Kac formula in two main ways. First, it
is extended
to nondifferentiable functions. Second, the process X is not required to
satisfy an SDE. Instead, it is only required to be a quasimartingale
satisfying
an integrability condition, and the martingale condition for f(t,X) is
then
expressed in terms of the marginal distributions, drift measure and
jumps of X.
The proof involves the stochastic calculus of Dirichlet processes and a
time-reversal argument.
These results are then applied to show that a continuous and strong
Markov
martingale is uniquely determined by its marginal distributions.
http://arxiv.org/abs/0803.3303
---------------------------------------------------------------
6829. ASYMPTOTICS OF INPUT-CONSTRAINED BINARY SYMMETRIC CHANNEL CAPACITY
Guangyue Han and Brian Marcus
In this paper, we study the classical problem of noisy constrained
capacity
in the case of the binary symmetric channel (BSC), namely, the
capacity of a
BSC whose input is a sequence from a constrained set. Motivated by a
result of
Ordentlich and Weissman, we derive an asymptotic formula (when the noise
parameter is small) for the entropy rate of a hidden Markov chain,
observed
when a Markov chain passes through a binary symmetric channel. Using
this
result we establish an asymptotic formula for the capacity of a binary
symmetric channel with input process supported on an irreducible
finite type
constraint, as the noise parameter tends to zero.
http://arxiv.org/abs/0803.3360
---------------------------------------------------------------
6830. MULTIVARIATE ANALYSIS AND JACOBI ENSEMBLES: LARGEST EIGENVALUE,
TRACY WIDOM LIMITS AND RATES OF CONVERGENCE
Iain M. Johnstone
Let $A$ and $B$ be independent, central Wishart matrices in $p$
variables
with common covariance and having $m$ and $n$ degrees of freedom
respectively.
The distribution of the largest eigenvalue of $(A+B)^{-1}B$ has numerous
applications in multivariate statistics, but is difficult to calculate
exactly.
Suppose that $m$ and $n$ grow in proportion to $p$. We show that after
centering and scaling, the distribution is approximated to second order,
$O(p^{-2/3})$, by the Tracy-Widom law. The results are obtained for both
complex and then real valued data by using methods of random matrix
theory to
study the largest eigenvalue of the Jacobi unitary and orthogonal
ensembles.
Asymptotic approximations of Jacobi polynomials near the largest zero
play a
central role.
http://arxiv.org/abs/0803.3408
---------------------------------------------------------------
6831. HOMOGENIZATION FOR SEMI-LINEAR PDE WITH DISCONTINUOUS COEFFICIENTS
K. Bahlali (IMATH) and Abouo Elouaflin (UFR-MI) and E. Pardoux (CMI)
We study the asymptotic behavior of the solution of semi-linear PDEs.
Neither
periodicity nor ergodicity assumptions are assumed. The coefficients
admit only
a limit in a Cesaro sense. In such a case, the limit coefficients may
have
discontinuity. We use probabilistic approach based on weak convergence
techniques for the associated backward stochastic differential
equation in the
S-topology. We establish weak continuity for the flow of the limit
diffusion
process and related the PDE limit to the backward stochastic
differential
equation via the representation of Lp-viscosity solution.
http://arxiv.org/abs/0803.3499
---------------------------------------------------------------
6832. MAXIMA OF DIRICHLET AND TRIANGULAR ARRAYS OF GAMMA VARIABLES
Arup Bose and Amites Dasgupta and Krishanu Maulik
Consider a rowwise independent triangular array of gamma random
variables
with varying parameters. Under several different conditions on the shape
parameter, we show that the sequence of row-maximums converges weakly
after
linear or power transformation. Depending on the parameter
combinations, we
obtain both Gumbel and non-Gumbel limits.
The weak limits for maximum of the coordinates of certain Dirichlet
vectors
of increasing dimension are also obtained using the gamma
representation.
http://arxiv.org/abs/0803.3518
---------------------------------------------------------------
6833. INTEGRATION WITH RESPECT TO LOCAL TIME AND ITO'S FORMULA FOR
SMOOTH NONDEGENERATE MARTINGALES
Xavier Bardina and Carles Rovira
We show an It\^ o's formula for nondegenerate Brownian martingales
$X_t=\int_0^t u_s dW_s$ and functions $F(x,t)$ with locally integrable
derivatives in $t$ and $x$. We prove that one can express the
additional term
in It\^o's s formula as an integral over space and time with respect
to local
time.
http://arxiv.org/abs/0803.3522
---------------------------------------------------------------
6834. ESCAPING THE BROWNIAN STALKERS
Alexander Weiss
We propose a simple model for the behaviour of longterm investors on a
stock
market, consisting of three particles, which represent the current
price of the
stock and the opinion of the buyers, respectively sellers, about the
right
trading price. As time evolves, both groups of traders update their
opinions
with respect to the current price. The update speed is controled by a
parameter
$\gamma$, the price process is described by a geometric Brownian
motion. We
consider the stability of the market in terms of the distance between
the
buyers' and sellers' opinion, and prove that the distance process is
recurrent/transient in dependence on $\gamma$.
http://arxiv.org/abs/0803.3590
---------------------------------------------------------------
6835. ON CONVERGENCE OF DYNAMICS OF HOPPING PARTICLES TO A BIRTH-AND-
DEATH PROCESS IN CONTINUUM
Dmitri Finkelshtein and Yuri Kondratiev and Eugene Lytvynov
We show that some classes of birth-and-death processes in continuum
(Glauber
dynamics) may be derived as a scaling limit of a dynamics of interacting
hopping particles (Kawasaki dynamics)
http://arxiv.org/abs/0803.3551
---------------------------------------------------------------
6836. THE EQUIVALENCE BETWEEN UNIQUENESS AND CONTINUOUS DEPENDENCE OF
SOLUTION FOR BSDES WITH CONTINUOUS COEFFICIENT
Guangyan Jia and Zhiyong Yu
In this paper, we will prove that, if the coefficient $g=g(t,y,z)$ of
a BSDE
is assumed to be continuous and linear growth in $(y,z)$, then the
uniqueness
of solution and continuous dependence with respect to $g$ and the
terminal
value $\xi$ are equivalent.
http://arxiv.org/abs/0803.3660
---------------------------------------------------------------
6837. INTEGRATION WITH RESPECT TO FRACTIONAL LOCAL TIMES WITH HURST
INDEX $1/2<H<1$
Litan Yan and Xiangfeng Yang
In this paper, we study the stochastic integration with respect to local
times of fractional Brownian motion $B^H$ with Hurst index $1/2<H<1$.
As a
related problem we define the {\it weighted quadratic covariation}
$$[f(B^H),B^H]^{(W)}$ of $f(B^H)$ and $B^H$ as follows $$
\sum_{k=0}^{n-1}
k^{2H-1}\{f(B^H_{t_{k+1}})-f(B^H_{t_{k}})\} (B^H_{t_{k+1}}-B^H_{t_{k}})
\stackrel{P} \longrightarrow [f(B^H),B^H]^{(W)}_t, $$ where $t_k=kt/n
$. By
applying the quadratic covariation we get an It\^o formula, and we
consider
also the process of the form $$\int_0^t f(B^H_s) dB^H_s
+2H[f(B^H),B^H]^{(W)}_t,
\qquad t\in [0,1].$$ These are also extended to the time-dependent case.
http://arxiv.org/abs/0803.3665
---------------------------------------------------------------
6838. THE GAME-THEORETIC MARTINGALES BEHIND THE ZERO-ONE LAWS
Akimichi Takemura and Vladimir Vovk and and Glenn Shafer
We prove game-theoretic generalizations of some well known zero-one
laws. Our
proofs make the martingales behind the laws explicit.
http://arxiv.org/abs/0803.3679
---------------------------------------------------------------
6839. NUMERICAL ALGORITHMS AND SIMULATIONS FOR REFLECTED BACKWARD
STOCHASTIC DIFFERENTIAL EQUATIONS WITH TWO CONTINUOUS BARRIERS
Mingyu Xu
In this paper we study different algorithms for reflected backward
stochastic
differential equations (BSDE in short) with two continuous barriers
basing on
random work framework. We introduce different numerical algorithms by
penalization method and reflected method. At last simulation results
are also
presented.
http://arxiv.org/abs/0803.3712
---------------------------------------------------------------
6840. ON DISTRIBUTIONAL PROPERTIES OF PERPETUITIES
Gerold Alsmeyer and Alex Iksanov and Uwe Roesler
We study probability distributions of convergent random series of a
special
structure, called perpetuities. By giving a new argument, we prove
that such
distributions are of pure type: degenerate, absolutely continuous, or
continuously singular. We further provide necessary and sufficient
criteria for
the finiteness of $p$-moments, $p>0$ as well as exponential moments. In
particular, a formula for the abscissa of convergence of the moment
generating
function is provided. The results are illustrated with a number of
examples at
the end of the article.
http://arxiv.org/abs/0803.3716
---------------------------------------------------------------
6841. NORMALLY REFLECTED BROWNIAN MOTION AND SPECTRAL PROPERTIES OF
THE NEUMANN LAPLACIAN IN UNBOUNDED DOMAINS
Ross Pinsky
Let $D\subsetneq R^d$ be an unbounded domain and let $B(t)$ be a
Brownian
motion in $D$ with normal reflection at the boundary, generated by
$\frac12\Delta_N$, where $\Delta_N$ is the Neumann Laplacian on $D$.
For a
bounded subdomain $U$, let $\tau_U=\inf\{t\ge0:B(t)\in \bar U\}$
denote the
first hitting time of $\bar U$. It is well-known that the behavior of
the
random variable $\tau_U$ determines the global behavior of the Brownian
motion--that is, its transience, null recurrence or positive
recurrence. In
fact, the behavior of $\tau_U$ as $U$ varies over all bounded subdomains
determines certain spectral properties of the Neumann Laplacian. In
this paper
we study the behavior of $\tau_U$ in order to treat both of these
issues. Most
of the work deals with domains of the form $D=\{(x,z)\in R^{l+m}:|z|
<H(|x|)\}$,
where $d=l+m$ and $H$ is a sufficiently regular function.
http://arxiv.org/abs/0803.3748
---------------------------------------------------------------
6842. THE FOURIER SPECTRUM OF CRITICAL PERCOLATION
Christophe Garban and Gabor Pete and Oded Schramm
Consider the indicator function $f$ of a two-dimensional percolation
crossing
event. In this paper, the Fourier transform of $f$ is studied and
sharp bounds
are obtained for its lower tail in several situations. Various
applications of
these bounds are derived. In particular, we show that the set of
exceptional
times of dynamical critical site percolation on the triangular grid in
which
the origin percolates has dimension 31/36 a.s., and the corresponding
dimension
in the half-plane is 5/9. It is also proved that critical bond
percolation on
the square grid has exceptional times a.s. Also, the asymptotics of
the number
of sites that need to be resampled in order to significantly perturb
the global
percolation configuration in a large square is determined.
http://arxiv.org/abs/0803.3750
---------------------------------------------------------------
6843. CONDITIONAL HAAR MEASURES ON CLASSICAL COMPACT GROUPS
Paul Bourgade
We give a probabilistic proof of the Weyl integration formula on the
unitary
group. This relies on a suitable definition of Haar measures
conditioned to the
existence of a stable subspace with any given dimension. The developed
method
leads to the following result : for this conditional measure, writing
$Z_U^{(p)}$ for the first non-zero derivative of the characteristic
polynomial
at 1, $Z_U^{(p)}$ can be decomposed as an explicit product of $n-p$
indepedent
random variables.
This implies a central limit theorem for $\log Z_U^{(p)}$ and
asymptotics for
the density of $Z_U^{(p)}$ near 0. Similar limit theorems are given
for the
orthogonal and symplectic groups.
http://arxiv.org/abs/0803.3753
---------------------------------------------------------------
6844. GEOMETRIC PROPERTIES OF TWO-DIMENSIONAL NEAR-CRITICAL PERCOLATION
Federico Camia and Matthijs Joosten and Ronald Meester
Using certain scaling relations for two-dimensional percolation, we
study
some global geometric properties of "near-critical" scaling limits. More
precisely, we show that when the lattice spacing is sent to zero,
there are
only three possible types of scaling limits for the collection of
percolation
interfaces: the trivial one consisting of no curves at all, the
critical one
corresponding to the full scaling limit of critical percolation, and
one in
which any deterministic point in the plane is surrounded with
probability one
by a largest loop and by a countably infinite family of nested loops
with radii
going to zero. All three cases occur. The first one corresponds to the
subcritical and supercritical phases. The last one corresponds to the
near-critical regime, with a scaling limit which is nontrivial, like the
critical one, but which, unlike the critical one, is not scale
invariant and
thus retains a flavor of supercritical percolation.
http://arxiv.org/abs/0803.3785
---------------------------------------------------------------
6845. THE REVERSE OF THE LAW OF LARGE NUMBERS
Kieran Kelly and Przemyslaw Repetowicz and Seosamh macReamoinn
The Law of Large Numbers tells us that as the sample size (N) is
increased,
the sample mean converges on the population mean, provided that the
latter
exists. In this paper, we investigate the opposite effect: keeping the
sample
size fixed while increasing the number of outcomes (M) available to a
discrete
random variable. We establish sufficient conditions for the variance
of the
sample mean to increase monotonically with the number of outcomes,
such that
the sample mean ``diverges'' from the population mean, acting like an
``reverse'' to the law of large numbers. These results, we believe, are
relevant to many situations which require sampling of statistics of
certain
finite discrete random variables.
http://arxiv.org/abs/0803.3913
---------------------------------------------------------------
6846. AN APPROXIMATION ALGORITHM FOR COUNTING CONTINGENCY TABLES
Alexander Barvinok and Zur Luria and Alex Samorodnitsky and and
Alexander Yong
We present a randomized approximation algorithm for counting contingency
tables, mxn non-negative integer matrices with given row sums
R=(r_1, ..., r_m)
and column sums C=(c_1, ..., c_n). We define smooth margins (R,C) in
terms of
the typical table and prove that for such margins the algorithm has
quasi-polynomial N^{O(ln N)} complexity, where N=r_1+...+r_m=c_1+...
+c_n.
Various classes of margins are smooth, e.g., when m=O(n), n=O(m) and
the ratios
between the largest and the smallest row sums as well as between the
largest
and the smallest column sums are strictly smaller than the golden ratio
(1+sqrt{5})/2 = 1.618. The algorithm builds on Monte Carlo integration
and
sampling algorithms for log-concave densities, the matrix scaling
algorithm,
the permanent approximation algorithm, and an integral representation
for the
number of contingency tables.
http://arxiv.org/abs/0803.3948
---------------------------------------------------------------
6847. THE TRACY--WIDOM LIMIT FOR THE LARGEST EIGENVALUES OF SINGULAR
COMPLEX WISHART MATRICES
Alexei Onatski
This paper extends the work of El Karoui [Ann. Probab. 35 (2007)
663--714]
which finds the Tracy--Widom limit for the largest eigenvalue of a
nonsingular
$p$-dimensional complex Wishart matrix $W_{\mathbb{C}}(\Omega_p,n)$ to
the case
of several of the largest eigenvalues of the possibly singular $(n<p)$
matrix
$W_{\mathbb{C}}(\Omega_p,n).$ As a byproduct, we extend all results of
Baik,
Ben Arous and Peche [Ann. Probab. 33 (2005) 1643--1697] to the
singular Wishart
matrix case. We apply our findings to obtain a 95% confidence set for
the
number of common risk factors in excess stock returns.
http://arxiv.org/abs/0803.4155
---------------------------------------------------------------
6848. SMALL DEVIATIONS OF SMOOTH STATIONARY GAUSSIAN PROCESSES
F.Aurzada and I.A.Ibragimov and M.A.Lifshits and J.H. van Zanten
We investigate the small deviation probabilities of a class of very
smooth
stationary Gaussian processes playing an important role in Bayesian
statistical
inference. Our calculations are based on the appropriate modification
of the
entropy method due to Kuelbs, Li, and Linde as well as on classical
results
about the entropy of classes of analytic functions. They also involve
Tsirelson's upper bound for small deviations and shed some light on
the limits
of sharpness for that estimate.
http://arxiv.org/abs/0803.4238
---------------------------------------------------------------
6849. ON SEQUENTIAL ESTIMATION AND PREDICTION FOR DISCRETE TIME SERIES
G. Morvai and B. Weiss
The problem of extracting as much information as possible from a
sequence of
observations of a stationary stochastic process $X_0,X_1,...X_n$ has
been
considered by many authors from different points of view. It has long
been
known through the work of D. Bailey that no universal estimator for
$\textbf{P}(X_{n+1}|X_0,X_1,...X_n)$ can be found which converges to
the true
estimator almost surely. Despite this result, for restricted classes of
processes, or for sequences of estimators along stopping times,
universal
estimators can be found. We present here a survey of some of the
recent work
that has been done along these lines.
http://arxiv.org/abs/0803.4332
---------------------------------------------------------------
6850. CONSISTENT PRICE SYSTEMS AND FACE-LIFTING PRICING UNDER
TRANSACTION COSTS
Paolo Guasoni and Mikl\'os R\'asonyi and Walter Schachermayer
In markets with transaction costs, consistent price systems play the
same
role as martingale measures in frictionless markets. We prove that if a
continuous price process has conditional full support, then it admits
consistent price systems for arbitrarily small transaction costs. This
result
applies to a large class of Markovian and non-Markovian models,
including
geometric fractional Brownian motion. Using the constructed price
systems, we
show, under very general assumptions, the following ``face-lifting''
result:
the asymptotic superreplication price of a European contingent claim
$g(S_T)$
equals $\hat{g}(S_0)$, where $\hat{g}$ is the concave envelope of $g$
and $S_t$
is the price of the asset at time $t$. This theorem generalizes
similar results
obtained for diffusion processes to processes with conditional full
support.
http://arxiv.org/abs/0803.4416
---------------------------------------------------------------
6851. STOCHASTIC SOLUTION OF A NONLINEAR FRACTIONAL DIFFERENTIAL
EQUATION
F. Cipriano and H. Ouerdiane and R. Vilela Mendes
A stochastic solution is constructed for a fractional generalization
of the
KPP (Kolmogorov, Petrovskii, Piskunov) equation. The solution uses a
fractional
generalization of the branching exponential process and propagation
processes
which are spectral integrals of Levy processes
http://arxiv.org/abs/0803.4457
---------------------------------------------------------------
6852. REAL AND COMPLEX ZEROS OF RIEMANNIAN RANDOM WAVES
Steve Zelditch
We consider Riemannian random waves, i.e. Gaussian random linear
combination
of eigenfunctions of the Laplacian on a compact Riemannian manifold with
frequencies from a short interval (`asymptotically fixed frequency').
We first
show that the expected limit distribution of the real zero set of a is
uniform
with respect to the volume form of a compact Riemannian manifold $(M,
g)$. We
then show that the complex zero set of the analytic continuations of
such
Riemannian random waves to a Grauert tube in the complexification of $M
$ tends
to a limit current.
http://arxiv.org/abs/0803.4334
---------------------------------------------------------------
6853. POISSON CLUSTER MEASURES: QUASI-INVARIANCE, INTEGRATION BY PARTS
AND EQUILIBRIUM STOCHASTIC DYNAMICS
Leonid Bogachev and Alexei Daletskii
The distribution $\mu_{cl}$ of a Poisson cluster process in $X=
\mathbb{R}^d$
(with i.i.d. clusters) is studied via an auxiliary Poisson measure on
the space
of configurations in $\mathfrak{X}=\sqcup_n X^n$, with intensity
defined as a
convolution of the background intensity of cluster centres and the
probability
distribution of a generic cluster. We show that the measure $\mu_{cl}$
is
quasi-invariant with respect to the group of compactly supported
diffeomorphisms of $X$ and prove an integration-by-parts formula for
$\mu_{cl}$. The corresponding equilibrium stochastic dynamics is then
constructed using the method of Dirichlet forms.
http://arxiv.org/abs/0803.4496
---------------------------------------------------------------
6854. A THEORETICAL STUDY OF MAFIA GAMES
Erlin Yao
Mafia can be described as an experiment in human psychology and mass
hysteria, or as a game between informed minority and uninformed
majority. Focus
on a very restricted setting, Mossel et al. [to appear in Ann. Appl.
Probab.
Volume 18, Number 2] showed that in the mafia game without detectives,
if the
civilians and mafias both adopt the optimal randomized strategy, then
the two
groups have comparable probabilities of winning exactly when the total
player
size is R and the mafia size is of order Sqrt(R). They also proposed a
conjecture which stated that this phenomenon should be valid in a more
extensive framework. In this paper, we first indicate that the main
theorem
given by Mossel et al. [to appear in Ann. Appl. Probab. Volume 18,
Number 2]
can not guarantee their conclusion, i.e., the two groups have comparable
winning probabilities when the mafia size is of order Sqrt(R). Then we
give a
theorem which validates the correctness of their conclusion. In the
last, by
proving the conjecture proposed by Mossel et al. [to appear in Ann.
Appl.
Probab. Volume 18, Number 2], we generalize the phenomenon to a more
extensive
framework, of which the mafia game without detectives is only a
special case.
http://arxiv.org/abs/0804.0071
---------------------------------------------------------------
6855. ESTIMATES OF TEMPERED STABLE DENSITIES
Pawe{\l} Sztonyk
Estimates of densities of convolution semigroups of probability
measures are
given under specific assumptions on the corresponding L\'evy measure
and the
L\'evy--Khinchin exponent. The assumptions are satisfied, e.g., by
tempered
stable semigroups of J. Rosi{\'n}ski.
http://arxiv.org/abs/0804.0113
---------------------------------------------------------------
6856. WEAK EXISTENCE OF THE SQUARED BESSEL PROCESS, CIR MODEL, AND
LONGSTAFF MODEL, WITH SKEW REFLECTION ON A DETERMINISTIC TIME
DEPENDENT CURVE
Gerald Trutnau
Using the technique of moving domains, and classical direct stochastic
calculus, we construct the Cox-Ingersoll-Ross process, as well as its
squareroot, with additional skew reflection on a deterministic time
dependent
curve.
http://arxiv.org/abs/0804.0119
---------------------------------------------------------------
6857. BASIC PROPERTIES OF NONLINEAR STOCHASTIC SCHR\"{O}DINGER
EQUATIONS DRIVEN BY BROWNIAN MOTIONS
Carlos M. Mora and Rolando Rebolledo
The paper is devoted to the study of nonlinear stochastic Schr
\"{o}dinger
equations driven by standard cylindrical Brownian motions (NSSEs)
arising from
the unraveling of quantum master equations. Under the Born--Markov
approximations, this class of stochastic evolutions equations on
Hilbert spaces
provides characterizations of both continuous quantum measurement
processes and
the evolution of quantum systems. First, we deal with the existence and
uniqueness of regular solutions to NSSEs. Second, we provide two general
criteria for the existence of regular invariant measures for NSSEs. We
apply
our results to a forced and damped quantum oscillator.
http://arxiv.org/abs/0804.0121
---------------------------------------------------------------
6858. PATHWISE UNIQUENESS OF THE SQUARED BESSEL PROCESS, AND CIR
PROCESS, WITH SKEW REFLECTION ON A DETERMINISTIC TIME DEPENDENT CURVE
Gerald Trutnau
We investigate pathwise uniqueness for the Cox-Ingersoll-Ross process
with
additional reflection term multiplied by some real integer between
zero and
one. The reflection term is the symmetric local time of the CIR
process at a
deterministic time dependent curve, which in general only needs to be
locally
of bounded variation.
http://arxiv.org/abs/0804.0123
---------------------------------------------------------------
6859. CONVEX PRICING BY A GENERALIZED ENTROPY PENALTY
Johannes Leitner
In an incomplete Brownian-motion market setting, we propose a convex
monotonic pricing functional for nonattainable bounded contingent
claims which
is compatible with prices for attainable claims. The pricing
functional is
defined as the convex conjugate of a generalized entropy penalty
functional and
an interpretation in terms of tracking with instantaneously vanishing
risk can
be given.
http://arxiv.org/abs/0804.0127
---------------------------------------------------------------
6860. PHASE TRANSITIONS IN PARTIALLY STRUCTURED RANDOM GRAPHS
Oskar Sandberg
We study a one parameter family of random graph models that spans a
continuum
between traditional random graphs of the Erd\H{o}s-R\'enyi type, where
there is
no underlying structure, and percolation models, where the possible
edges are
dictated exactly by a geometry. We find that previously developed
theories in
the fields of random graphs and percolation have, starting from
different
directions, covered almost all the models described by our family. In
particular, the existence or not of a phase transition where a giant
cluster
arises has been proved for all values of the parameter but one. We
prove that
the single remaining case behaves like a random graph and has a single
linearly
sized cluster when the expected vertex degree is greater than one.
http://arxiv.org/abs/0804.0137
---------------------------------------------------------------
6861. RENORMALIZATION OF THE TWO-DIMENSIONAL LOTKA--VOLTERRA MODEL
J. Theodore Cox and Edwin A. Perkins
We show that renormalized two-dimensional Lotka--Volterra models near
criticality converge to a super-Brownian motion. This is used to
establish
long-term survival of a rare type for a range of parameter values near
the
voter model.
http://arxiv.org/abs/0804.0158
---------------------------------------------------------------
6862. ESTIMATING CORRELATION FROM HIGH, LOW, OPENING AND CLOSING PRICES
L. C. G. Rogers and Fanyin Zhou
In earlier studies, the estimation of the volatility of a stock using
information on the daily opening, closing, high and low prices has been
developed; the additional information in the high and low prices can be
incorporated to produce unbiased (or near-unbiased) estimators with
substantially lower variance than the simple open--close estimator.
This paper
tackles the more difficult task of estimating the correlation of two
stocks
based on the daily opening, closing, high and low prices of each. If
we had
access to the high and low values of some linear combination of the
two log
prices, then we could use the univariate results via polarization, but
this is
not data that is available. The actual problem is more challenging; we
present
an unbiased estimator which halves the variance.
http://arxiv.org/abs/0804.0162
---------------------------------------------------------------
6863. CHARACTERIZATION OF THE CRITICAL VALUES OF BRANCHING RANDOM
WALKS ON WEIGHTED GRAPHS THROUGH INFINITE-TYPE BRANCHING PROCESSES
Daniela Bertacchi and Fabio Zucca
We study the branching random walk on weighted graphs; site-breeding and
edge-breeding branching random walks on graphs are seen as particular
cases. We
describe the strong critical value in terms of a geometrical parameter
of the
graph. We characterize the weak critical value and relate it to another
geometrical parameter. We prove that, at the strong critical value,
the process
dies out locally almost surely; while, at the weak critical value,
global
survival and global extinction are both possible.
http://arxiv.org/abs/0804.0224
---------------------------------------------------------------
6864. REAL ROOTS OF RANDOM POLYNOMIALS AND ZERO CROSSING PROPERTIES
OF DIFFUSION EQUATION
Gregory Schehr and Satya N. Majumdar
We study various statistical properties of real roots of three different
classes of random polynomials which recently attracted a vivid
interest in the
context of probability theory and quantum chaos. We first focus on gap
probabilities on the real axis, i.e. the probability that these
polynomials
have no real root in a given interval. For generalized Kac
polynomials, indexed
by an integer d, of large degree n, one finds that the probability of
no real
root in the interval [0,1] decays as a power law n^{-\theta(d)} where
\theta(d)
> 0 is the persistence exponent of the diffusion equation with random
initial
conditions in spatial dimension d. For n \gg 1 even, the probability
that they
have no real root on the full real axis decays like
n^{-2(\theta(2)+\theta(d))}. For Weyl polynomials and Binomial
polynomials,
this probability decays respectively like \exp{(-2\theta_{\infty}}
\sqrt{n})
and \exp{(-\pi \theta_{\infty} \sqrt{n})} where \theta_{\infty} is
such that
\theta(d) = 2^{-3/2} \theta_{\infty} \sqrt{d} in large dimension d. We
also
show that the probability that such polynomials have exactly k roots
on a given
interval [a,b] has a scaling form given by \exp{(-N_{ab} \tilde
\phi(k/N_{ab}))} where N_{ab} is the mean number of real roots in
[a,b] and
\tilde \phi(x) a universal scaling function. We develop a simple Mean
Field
(MF) theory reproducing qualitatively these scaling behaviors, and
improve
systematically this MF approach using the method of persistence with
partial
survival, which in some cases yields exact results. Finally, we show
that the
probability density function of the largest absolute value of the real
roots
has a universal algebraic tail with exponent {-2}. These analytical
results are
confirmed by detailed numerical computations.
http://arxiv.org/abs/0803.4396
---------------------------------------------------------------
6865. QUENCHED LARGE DEVIATIONS FOR RANDOM WALK IN A RANDOM ENVIRONMENT
Atilla Yilmaz
We take the point of view of a particle performing random walk with
bounded
jumps on Z^d in a stationary and ergodic random environment. We prove
the
quenched large deviation principle (LDP) for the pair empirical
measure of the
environment Markov chain. By the contraction principle, we deduce the
quenched
LDP for the mean velocity of the particle and obtain a variational
formula for
the corresponding rate function. We propose an Ansatz for the
minimizer of this
formula. We verify this Ansatz for nearest-neighbor walks on Z. As a
separate
result, we give a probabilistic formula for the ergodic invariant
density of
the environment Markov chain in the case of ballistic random walk with
bounded
jumps on Z.
http://arxiv.org/abs/0804.0262
---------------------------------------------------------------
6866. ITO'S FORMULA IN UMD BANACH SPACES AND REGULARITY OF SOLUTIONS
OF THE ZAKAI EQUATION
Z. Brzezniak and J. M. A. M. van Neerven and M. C. Veraar and L. Weis
Using the theory of stochastic integration for processes with values
in a UMD
Banach space developed recently by the authors, an Ito formula is
proved which
is applied to prove the existence of strong solutions for a class of
stochastic
evolution equations in UMD Banach spaces. The abstract results are
applied to
prove regularity in space and time of the solutions of the Zakai
equation.
http://arxiv.org/abs/0804.0302
---------------------------------------------------------------
6867. EXISTENCE OF AN INFINITE PARTICLE LIMIT OF STOCHASTIC RANKING
PROCESS
Kumiko Hattori and Tetsuya Hattori
We study a stochastic particle system which models the time evolution
of the
ranking of books by online bookstores (e.g., Amazon). In this system,
particles
are lined in a queue. Each particle jumps at random jump times to the
top of
the queue, and otherwise stays in the queue, being pushed toward the
tail every
time another particle jumps to the top. In an infinite particle limit,
the
random motion of each particle between its jumps converges to a
deterministic
trajectory. (This trajectory is actually observed in the ranking data
on web
sites.) We prove that the (random) empirical distribution of this
particle
system converges to a deterministic space-time dependent distribution.
A core
of the proof is the law of large numbers for {\it dependent} random
variables.
http://arxiv.org/abs/0804.0321
---------------------------------------------------------------
6868. ASYMPTOTIC PROPERTIES OF THE MAXIMUM LIKELIHOOD ESTIMATOR FOR
STOCHASTIC PARABOLIC EQUATIONS WITH ADDITIVE FRACTIONAL BROWNIAN MOTION
Igor Cialenco and Sergey Lototsky and Jan Pospisil
A parameter estimation problem is considered for a diagonaliazable
stochastic
evolution equation using a finite number of the Fourier coefficients
of the
solution. The equation is driven by additive noise that is white in
space and
fractional in time with the Hurst parameter $H\geq 1/2$.
The objective is to study asymptotic properties of the maximum
likelihood
estimator as the number of the Fourier coefficients increases. A
necessary and
sufficient condition for consistency and asymptotic normality is
presented in
terms of the eigenvalues of the operators in the equation.
http://arxiv.org/abs/0804.0407
---------------------------------------------------------------
6869. PROBABILISTIC INTERPRETATION FOR SYSTEMS OF ISAACS EQUATIONS
WITH TWO REFLECTING BARRIERS
Rainer Buckdahn and Juan Li
In this paper we investigate zero-sum two-player stochastic differential
games whose cost functionals are given by doubly controlled reflected
backward
stochastic differential equations (RBSDEs) with two barriers. For
admissible
controls which can depend on the whole past and so include, in
particular,
information occurring before the beginning of the game, the games are
interpreted as games of the type "admissible strategy" against
"admissible
control", and the associated lower and upper value functions are
studied. A
priori random, they are shown to be deterministic, and it is proved
that they
are the unique viscosity solutions of the associated upper and the lower
Bellman-Isaacs equations with two barriers, respectively. For the
proofs we
make full use of the penalization method for RBSDEs with one barrier
and RBSDEs
with two barriers. For this end we also prove new estimates for RBSDEs
with two
barriers, which are sharper than those in [18]. Furthermore, we show
that the
viscosity solution of the Isaacs equation with two reflecting barriers
not only
can be approximated by the viscosity solutions of penalized Isaacs
equations
with one barrier, but also directly by the viscosity solutions of
penalized
Isaacs equations without barrier.
http://arxiv.org/abs/0804.0311
---------------------------------------------------------------
6870. HYPERDETERMINANTAL POINT PROCESSES
Steven N. Evans and Alex Gottlieb
As well as arising naturally in the study of non-intersecting random
paths,
random spanning trees, and eigenvalues of random matrices,
determinantal point
processes (sometimes also called fermionic point processes) are
relatively easy
to simulate and provide a quite broad class of models that exhibit
repulsion
between points. The fundamental ingredient used to construct a
determinantal
point process is a kernel giving the pairwise interactions between
points: the
joint distribution of any number of points then has a simple
expression in
terms of determinants of certain matrices defined from this kernel. In
this
paper we initiate the study of an analogous class of point processes
that are
defined in terms of a kernel giving the interaction between $2M$
points for
some integer $M$. The role of matrices is now played by $2M$-dimensional
"hypercubic" arrays, and the determinant is replaced by a suitable
generalization of it to such arrays -- Cayley's first
hyperdeterminant. We show
that some of the desirable features of determinantal point processes
continue
to be exhibited by this generalization.
http://arxiv.org/abs/0804.0450
---------------------------------------------------------------
6871. AN INTRODUCTION TO L\'{E}VY PROCESSES WITH APPLICATIONS IN FINANCE
Antonis Papapantoleon
These lectures notes aim at introducing L\'{e}vy processes in an
informal and
intuitive way, accessible to non-specialists in the field. In the
first part,
we focus on the theory of L\'{e}vy processes. We analyze a `toy'
example of a
L\'{e}vy process, viz. a L\'{e}vy jump-diffusion, which yet offers
significant
insight into the distributional and path structure of a L\'{e}vy
process. Then,
we present several important results about L\'{e}vy processes, such as
infinite
divisibility and the L\'{e}vy-Khintchine formula, the L\'{e}vy-It\^{o}
decomposition, the It\^{o} formula for L\'{e}vy processes and Girsanov's
transformation. Some (sketches of) proofs are presented, still the
majority of
proofs is omitted and the reader is referred to textbooks instead. In
the
second part, we turn our attention to the applications of L\'{e}vy
processes in
financial modeling and option pricing. We discuss how the price
process of an
asset can be modeled using L\'{e}vy processes and give a brief account
of
market incompleteness. Popular models in the literature are presented
and
revisited from the point of view of L\'{e}vy processes, and we also
discuss
three methods for pricing financial derivatives. Finally, some
indicative
evidence from applications to market data is presented.
http://arxiv.org/abs/0804.0482
---------------------------------------------------------------
6872. CONVERGENCE PROPERTIES OF KEMP'S Q-BINOMIAL DISTRIBUTION
Stefan Gerhold and Martin Zeiner
We consider Kemp's q-analogue of the binomial distribution. Several
convergence results involving the classical binomial, the Heine, the
discrete
normal, and the Poisson distribution are established. Some of them are
q-analogues of classical convergence properties. Besides elementary
estimates,
we apply Mellin transform asymptotics.
http://arxiv.org/abs/0804.0534
---------------------------------------------------------------
6873. DECENTRALIZED SEARCH WITH RANDOM COSTS
Oskar Sandberg
A decentralized search algorithm is a method of routing on a random
graph
that uses only limited, local, information about the realization of
the graph.
In some random graph models it is possible to define such algorithms
which
produce short paths when routing from any vertex to any other, while
for others
it is not.
We consider random graphs with random costs assigned to the edges.
In this
situation, we use the methods of stochastic dynamic programming to
create a
decentralized search method which attempts to minimize the total cost,
rather
than the number of steps, of each path. We show that it succeeds in
doing so
among all decentralized search algorithms which monotonically approach
the
destination. Our algorithm depends on knowing the expected cost of
routing from
every vertex to any other, but we show that this may be calculated
iteratively,
and in practice can be easily estimated from the cost of previous
routes and
compressed into a small routing table. The methods applied here can
also be
applied directly in other situations, such as efficient searching in
graphs
with varying vertex degrees.
http://arxiv.org/abs/0804.0577
---------------------------------------------------------------
6874. ON THE ROLE OF CONVEXITY IN FUNCTIONAL AND ISOPERIMETRIC
INEQUALITIES
Emanuel Milman
This is a continuation of our previous work 0712.4092. It is well
known that
various isoperimetric inequalities imply their functional
``counterparts'', but
in general this is not an equivalence. We show that under certain
convexity
assumptions (e.g. for log-concave probability measures in Euclidean
space), the
latter implication can in fact be reversed for very general
inequalities,
generalizing a reverse form of Cheeger's inequality due to Buser and
Ledoux. We
develop a coherent single framework for passing between isoperimetric
inequalities, Orlicz-Sobolev functional inequalities and capacity
inequalities,
the latter being notions introduced by Maz'ya and extended by
Barthe--Cattiaux--Roberto. As an application, we extend the known
results due
to the latter authors about the stability of the isoperimetric profile
under
tensorization, when there is no Central-Limit obstruction. As another
application, we show that under our convexity assumptions, $q$-log-
Sobolev
inequalities ($q \in [1,2]$) are equivalent to an appropriate family of
isoperimetric inequalities, extending results of Bakry--Ledoux and
Bobkov--Zegarlinski. Our results extend to the more general setting of
Riemannian manifolds with density which satisfy the $CD(0,\infty)$
curvature-dimension condition of Bakry--\'Emery.
http://arxiv.org/abs/0804.0453
---------------------------------------------------------------
6875. THE STRUCTURE OF UNICELLULAR MAPS, AND A CONNECTION BETWEEN MAPS
OF POSITIVE GENUS AND PLANAR LABELLED TREES
Guillaume Chapuy
A unicellular map is a map which has only one face. We give a bijection
between a dominant subset of rooted unicellular maps of fixed genus
and a set
of rooted plane trees with distinguished vertices. The bijection
applies as
well to the case of labelled unicellular maps, which are related to
all rooted
maps by Marcus and Schaeffer's bijection.
This gives an immediate derivation of the asymptotic number of
unicellular
maps of given genus, and a simple bijective proof of a formula of
Lehman and
Walsh on the number of triangulations with one vertex. From the
labelled case,
we deduce an expression of the asymptotic number of maps of genus g
with n
edges involving the ISE random measure, and an explicit
characterization of the
limiting profile and radius of random bipartite quadrangulations of
genus g in
terms of the ISE.
http://arxiv.org/abs/0804.0546
---------------------------------------------------------------
6876. A SYSTEM OF GRABBING PARTICLES RELATED TO GALTON-WATSON TREES
Jean Bertoin (DMA and PMA) and Vladas Sidoravicius (UCI and CWI)
and Maria Eulalia Vares (CBPF)
We consider a system of particles with arms that are activated
randomly to
grab other particles as a toy model for polymerization. We assume that
the
following two rules are fulfilled: Once a particle has been grabbed
then it
cannot be grabbed again, and an arm cannot grab a particle that
belongs to its
own cluster. We are interested in the shape of a typical polymer in the
situation when the initial number of monomers is large and the numbers
of arms
of monomers are given by i.i.d. random variables. Our main result is a
limit
theorem for the empirical distribution of polymers, where limit is
expressed in
terms of a Galton-Watson tree.
http://arxiv.org/abs/0804.0726
---------------------------------------------------------------
6877. CRYPTANALYSIS OF THE ALGEBRAIC ERASER AND SHORT EXPRESSIONS OF
PERMUTATIONS AS PRODUCTS
Arkadius Kalka and Mina Teicher and and Boaz Tsaban
On March 2004, Anshel, Anshel, Goldfeld, and Lemieux introduced the
_Algebraic Eraser_ scheme for key agreement over an insecure channel.
This
scheme is based on semidirect products of algebraic structures, and
uses a
novel hybrid of infinite and finite noncommutative groups. They also
introduced
the_Colored Burau Key Agreement Protocol (CBKAP)_, a concrete
realization of
this scheme. We present an efficient method to extract the shared key
out of
the public information provided by CBKAP, assuming that the keys are
chosen
with standard distributions.
Our methods come from probabilistic group theory, and seem to have
not been
used before in cryptanalysis. Of independent interest may be a simple
heuristic
algorithm we propose for finding short expressions of permutations as
products
of given random permutations. According to heuristic analysis
supported by
experiments, our algorithm gives expressions of length O(n^2log n) in
running
time O(n^4log n).
http://arxiv.org/abs/0804.0629
---------------------------------------------------------------
6878. COMPLEMENTS AND SIGNED DIGIT REPRESENTATIONS: ANALYSIS OF A
MULTI-EXPONENTIATION-ALGORITHM OF WU, LOU, LAI AND CHANG
Clemens Heuberger and Helmut Prodinger
Wu, Lou, Lai and Chang proposed a multi-exponentiation algorithm using
binary
complements and the non-adjacent form. The purpose of this paper is to
show
that neither the analysis of the algorithm given by its original
proposers nor
that by other authors are correct. In fact it turns out that the
complement
operation does not have significant influence on the performance of the
algorithm and can therefore be omitted.
http://arxiv.org/abs/0804.0733
---------------------------------------------------------------
6879. MARKOV JUMP PROCESSES APPROXIMATING A NONSYMMETRIC GENERALIZED
DIFFUSION
Nedzad Limi\'c
Consider a nonsymetric generalized diffusion $X(\cdot)$ in ${\bbR}^d$
generated by the differential operator $A(\msx)=\sum_{ij}
\partial_ia_{ij}(\msx)\partial_j +\sum_i b_i(\msx)\partial_i$. In this
paper
the diffusion process is approximated by Markov jump processes
$X_n(\cdot)$ in
homogeneous and isotropic grids $G_n \subset {\bbR}^d$ which converge in
distribution to diffusion. The generators of $X_n(\cdot)$ are
constructed
explicitly. Due to the homogeneity and isotropy of grids the proposed
method
for $d\geq3$ can be applied to processes for which the diffusion tensor
$\{a_{ij}(\msx)\}_{11}^{dd}$ fulfills an additional condition. The
proposed
construction offers a simple method for simulation of sample paths of
nonsymetric generalized diffusion. Simulations are carried out in
terms of jump
processes $X_n(\cdot)$. For $d=2$ the construction can be easily
implemented
into a computer code.
http://arxiv.org/abs/0804.0848
---------------------------------------------------------------
6880. LIMIT THEOREMS FOR SUBCRITICAL BRANCHING PROCESS IN RANDOM
ENVIRONMENT DEPENDING ON THE INITIAL NUMBER OF PARTICLES
Vincent Bansaye (PMA)
Asymptotic behaviors for subcritical Branching Processes in Random
Environment (BPRE) starting with several particles depend on whether
the BPRE
is strongly subcritical (SS), intermediate subcritical (IS) or weakly
subcritical (WS) (see \cite{bpree}). Descendances of particles for
BPRE are not
independent. In the (SS+IS) case, the asymptotic probability of
survival is
proportional to the initial number of particles. And conditionally on
the
survival of the population, only one initial particle survives a.s.
These two
properties do not hold in the (WS) case and different asymptotics are
established, which require to prove new results on random walk with
negative
drift. We provide an interpretation of these results by characterizing
the
sequence of environments selected when we condition by the survival of
particles. This also raises the problem of the dependence of the Yaglom
quasistationary distributions on the initial number of particles and the
asymptotic behavior of the Q-process associated with a subcritical BPRE.
http://arxiv.org/abs/0804.0853
---------------------------------------------------------------
6881. CLIQUE PERCOLATION
Bela Bollobas and Oliver Riordan
Derenyi, Palla and Vicsek introduced the following dependent percolation
model, in the context of finding communities in networks. Starting
with a
random graph $G$ generated by some rule, form an auxiliary graph $G'$
whose
vertices are the $k$-cliques of $G$, in which two vertices are joined
if the
corresponding cliques share $k-1$ vertices. They considered in
particular the
case where $G=G(n,p)$, and found heuristically the threshold for a giant
component to appear in $G'$. Here we give a rigorous proof of this
result, as
well as many extensions. The model turns out to be very interesting
due to the
essential global dependence present in $G'$.
http://arxiv.org/abs/0804.0867
---------------------------------------------------------------
6882. ON GAUSSIAN BRUNN-MINKOWSKI INEQUALITIES
Franck Barthe (IMT) and Nolwen Huet (IMT)
In this paper, we are interested in Gaussian versions of the classical
Brunn-Minkowski inequality. We prove in a streamlined way a semigroup
version
of the Ehrard inequality for $m$ Borel or convex sets based on a
previous work
by Borell. Our method allows us also to have semigroup proofs of the
classical
Brascamp-Lieb inequality and of the reverse one which follow exactly
the same
lines.
http://arxiv.org/abs/0804.0886
---------------------------------------------------------------
6883. SPIKED MODELS IN WISHART ENSEMBLE
Dong Wang
The spiked model is an important special case of the Wishart ensemble,
and a
natural generalization of the white Wishart ensemble. Mathematically,
it can be
defined on three kinds of variables: the real, the complex and the
quaternion.
For practical application, we are interested in the limiting
distribution of
the largest sample eigenvalue.
We first give a new proof of the result of Baik, Ben Arous and P
\'{e}ch\'{e}
for the complex spiked model, based on the method of multiple orthogonal
polynomials by Bleher and Kuijlaars. Then in the same spirit we
present a new
result of the rank 1 quaternionic spiked model, proven by combinatorial
identities involving quaternionic Zonal polynomials (\alpha = 1/2 Jack
polynomials) and skew orthogonal polynomials.
We find a phase transition phenomenon for the limiting distribution
in the
rank 1 quaternionic spiked model as the spiked population eigenvalue
increases,
and recognize the seemingly new limiting distribution on the critical
point as
the limiting distribution of the largest sample eigenvalue in the real
white
Wishart ensemble.
Finally we give conjectures for higher rank quaternionic spiked
model and the
real spiked model.
http://arxiv.org/abs/0804.0889
---------------------------------------------------------------
6884. A LOG-TYPE MOMENT RESULT FOR PERPETUITIES AND ITS APPLICATION
TO MARTINGALES IN SUPERCRITICAL BRANCHING RANDOM WALKS
Gerold Alsmeyer and Alexander Iksanov
Infinite sums of i.i.d. random variables discounted by a multiplicative
random walk are called perpetuities and have been studied by many
authors. The
present paper provides a log-type moment result for such random
variables under
minimal conditions which is then utilized for the study of related
moments of
a.s. limits of certain martingales associated with the supercritical
branching
random walk. The connection, first observed by the second author in
[Iksanov,
A.M. (2004). Elementary fixed points of the BRW smoothing transforms
with
infinite number of summands. Stoch. Proc. Appl. 114, 27-50.], arises
upon
consideration of a size-biased version of the branching random walk
originally
introduced by Lyons in [Lyons, R.(1997). A simple path to Biggins'
martingale
convergence for branching random walk. In Athreya, K.B., Jagers, P.
(eds.).
Classical and Modern Branching Processes, IMA Volumes in Mathematics
and its
Applications, vol. 84, Springer, Berlin, 217-221.]. We also provide a
necessary
and sufficient condition for uniform integrability of these
martingales in the
most general situation which particularly means that the classical
(LlogL)-condition is not always needed.
http://arxiv.org/abs/0804.0961
---------------------------------------------------------------
6885. LARGE DEVIATIONS PRINCIPLE FOR PERTURBED CONSERVATION LAWS
Mauro Mariani
We investigate large deviations for a family of conservative
stochastic PDEs
(conservation laws) in the asymptotic of jointly vanishing noise and
viscosity.
We obtain a first large deviations principle in a space of Young
measures. The
associated rate functional vanishes on a wide set, the so-called set of
measure-valued solutions to the limiting conservation law. We therefore
investigate a second order large deviations principle, thus providing a
quantitative characterization of non-entropic solutions to the
conservation
law.
http://arxiv.org/abs/0804.0997
---------------------------------------------------------------
6886. PRUNING A L\'EVY CONTINUUM RANDOM TREE
Romain Abraham (MAPMO) and Jean-Francois Delmas (CERMICS) and
Guillaume Voisin (MAPMO)
Given a general critical or sub-critical branching mechanism, we
define a
pruning procedure of the associated L\'evy continuum random tree. This
pruning
procedure is defined by adding some marks on the tree, using L\'evy
snake
techniques. We then prove that the resulting sub-tree after pruning is
still a
L\'evy continuum random tree. This last result is proved using the
exploration
process that codes the CRT, a special Markov property and martingale
problems
for exploration processes. We finally give the joint law under the
excursion
measure of the lengths of the excursions of the initial exploration
process and
the pruned one.
http://arxiv.org/abs/0804.1027
---------------------------------------------------------------
6887. MULTIVARIATE FELLER CONDITIONS IN TERM STRUCTURE MODELS: WHY
DO(N'T) WE CARE?
Peter Spreij and Enno Veerman and Peter Vlaar
In this paper, the relevance of the Feller conditions in discrete time
macro-finance term structure models is investigated. The Feller
conditions are
usually imposed on a continuous time multivariate square root process
to ensure
that the roots have nonnegative arguments. For a discrete time
approximate
model, the Feller conditions do not give this guarantee. Moreover, in a
macro-finance context the restrictions imposed might be economically
unappealing. At the same time, it has also been observed that even
without the
Feller conditions imposed, for a practically relevant term structure
model,
negative arguments rarely occur. Using models estimated on German
data, we
compare the yields implied by (approximate) analytic exponentially
affine
expressions to those obtained through Monte Carlo simulations of very
high
numbers of sample paths. It turns out that the differences are rarely
statistically significant, whether the Feller conditions are imposed
or not.
Moreover, economically the differences are negligible, as they are
always below
one basis point.
http://arxiv.org/abs/0804.1039
---------------------------------------------------------------
6888. A FRACTIONAL POISSON EQUATION: EXISTENCE, REGULARITY AND
APPROXIMATIONS
Marta Sanz-Sol\'e and Iv\'an Torrecilla-Tarantino
We consider a stochastic boundary value elliptic problem on a bounded
domain
$D\subset \mathbb{R}^k$, driven by a fractional Brownian field with
Hurst
parameter $H=(H_1,...,H_k)\in[{1/2},1[^k$. First we define the
stochastic
convolution derived from the Green kernel and prove some properties.
Using
monotonicity methods, we prove existence and uniqueness of solution,
along with
regularity of the sample paths. Finally, we propose a sequence of
lattice
approximations and prove its convergence to the solution of the SPDE
at a given
rate.
http://arxiv.org/abs/0804.1108
---------------------------------------------------------------
6889. STOCHASTIC EVOLUTION EQUATIONS IN UMD BANACH SPACES
J. M. A. M. van Neerven and M. C. Veraar and L. Weis
We discuss existence, uniqueness, and space-time H\"older regularity for
solutions of the parabolic stochastic evolution equation dU(t) =
(AU(t) +
F(t,U(t))) dt + B(t,U(t)) dW_H(t), t\in [0,\Tend], U(0) = u_0, where $A$
generates an analytic $C_0$-semigroup on a UMD Banach space $E$ and
$W_H$ is a
cylindrical Brownian motion with values in a Hilbert space $H$. We
prove that
if the mappings $F:[0,T]\times E\to E$ and $B:[0,T]\times E\to
\mathscr{L}(H,E)$ satisfy suitable Lipschitz conditions and $u_0$ is
$\F_0$-measurable and bounded, then this problem has a unique mild
solution,
which has trajectories in $C^\l([0,T];\D((-A)^\theta)$ provided $
\lambda\ge 0$
and $\theta\ge 0$ satisfy $\l+\theta<\frac12$. Various extensions of
this
result are given and the results are applied to parabolic stochastic
partial
differential equations.
http://arxiv.org/abs/0804.0932
---------------------------------------------------------------
6890. ANALYSIS OF DISCRETE AND HYBRID STOCHASTIC SYSTEMS BY NONLINEAR
CONTRACTION THEORY
Quang-Cuong Pham
We investigate the stability properties of discrete and hybrid
stochastic
nonlinear dynamical systems. More precisely, we extend the stochastic
contraction theorems (which were formulated for continuous systems) to
the case
of discrete and hybrid resetting systems. In particular, we show that
the mean
square distance between any two trajectories of a discrete (or hybrid
resetting) contracting stochastic system is upper-bounded by a
constant after
exponential transients. Using these results, we study the
synchronization of
noisy nonlinear oscillators coupled by discrete noisy interactions.
http://arxiv.org/abs/0804.0934
---------------------------------------------------------------
6891. A NEW ESTIMATOR FOR THE NUMBER OF SPECIES IN A POPULATION
L. Cecconi and A. Gandolfi and C. C. A. Sastri
We consider the classic problem of estimating T, the total number of
species
in a population, from repeated counts in a simple random sample. We
look first
at the Chao-Lee estimator: we initially show that such estimator can be
obtained by reconciling two estimators of the unobserved probability,
and then
develop a sequence of improvements culminating in a Dirichlet prior
Bayesian
reinterpretation of the estimation problem. By means of this, we obtain
simultaneous estimates of T, of the normalized interspecies variance $
\gamma^2$
and of the parameter $\lambda$ of the prior. Several simulations show
that our
estimation method is more flexible than several known methods we used as
comparison; the only limitation, apparently shared by all other
methods, seems
to be that it cannot deal with the rare cases in which $\gamma^2 >1$
http://arxiv.org/abs/0804.1030
---------------------------------------------------------------
6892. THE COMPOUND POISSON DISTRIBUTION AND RETURN TIMES IN DYNAMICAL
SYSTEMS
N. Haydn and S. Vaienti
Previously it has been shown that some classes of mixing dynamical
systems
have limiting return times distributions that are almost everywhere
Poissonian.
Here we study the behaviour of return times at periodic points and
show that
the limiting distribution is a compound Poissonian distribution. We
also derive
error terms for the convergence to the limiting distribution. We also
prove a
very general theorem that can be used to establish compound Poisson
distributions in many other settings.
http://arxiv.org/abs/0804.1032
---------------------------------------------------------------
6893. BRANCHING PROCESSES IN RANDOM ENVIRONMENT DIE SLOWLY
V.Vatutin and A.E.Kyprianou
Let $Z_{n,}n=0,1,...,$ be a branching process evolving in the random
environment generated by a sequence of iid generating functions $%
f_{0}(s),f_{1}(s),...,$ and let $S_{0}=0,S_{k}=X_{1}+...+X_{k},k\geq
1,$ be the
associated random walk with $X_{i}=\log f_{i-1}^{\prime}(1),$ $\tau
(m,n)$ be
the left-most point of minimum of $\left\{S_{k},k\geq 0\right\} $ on the
interval $[m,n],$ and $T=\min \left\{k:Z_{k}=0\right\} $. Assuming
that the
associated random walk satisfies the Doney condition $P(S_{n}>0) \to
\rho \in
(0,1),n\to \infty ,$ we prove (under the quenched approach)
conditional limit
theorems, as $n\to \infty $, for the distribution of $Z_{nt},$ $Z_{\tau
(0,nt)},$ and $Z_{\tau (nt,n)},$ $t\in (0,1),$ given $T=n$. It is
shown that
the form of the limit distributions essentially depends on the
location of
$\tau (0,n)$ with respect to the point $nt.$
http://arxiv.org/abs/0804.1155
---------------------------------------------------------------
6894. CHAINING TECHNIQUES AND THEIR APPLICATION TO STOCHASTIC FLOWS
Michael Scheutzow
We review several competing chaining methods to estimate the supremum,
the
diameter of the range or the modulus of continuity of a stochastic
process in
terms of tail bounds of their two-dimensional distributions. Then we
show how
they can be applied to obtain upper bounds for the growth of bounded
sets under
the action of a stochastic flow.
http://arxiv.org/abs/0804.1263
---------------------------------------------------------------
6895. THE EXPANSION FOR THE OVERLAP FUNCTION
Sergio De Carvalho Bezerra
In this work, it is proved the complete expansion for the second
moment of
the overlap function for the Sherrington-Kirkpatrick model. It is a
technical
result which takes advantage of the cavity method and other induction
arguments.
http://arxiv.org/abs/0804.1339
---------------------------------------------------------------
6896. WEAK APPROXIMATION OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS:
THE NON LINEAR CASE
Arnaud Debussche (IRMAR)
We study the error of the Euler scheme applied to a stochastic partial
differential equation. We prove that as it is often the case, the weak
order of
convergence is twice the strong order. A key ingredient in our proof is
Malliavin calculus which enables us to get rid of the irregular terms
of the
error. We apply our method to the case a semilinear stochastic heat
equation
driven by a space-time white noise.
http://arxiv.org/abs/0804.1304
---------------------------------------------------------------
6897. A FREDHOLM DETERMINANT REPRESENTATION IN ASEP
Craig A. Tracy and Harold Widom
In previous work the authors found integral formulas for probabilities
in the
asymmetric simple exclusion process (ASEP) on the integer lattice. The
dynamics
are uniquely determined once the initial state is specified. In this
note we
restrict our attention to the case of step initial condition with
particles at
the positive integers, and consider the distribution function for the
m'th
particle from the left. In the previous work an infinite series of
multiple
integrals was derived for this distribution. In this note we show that
the
series can be summed to give a single integral whose integrand
involves a
Fredholm determinant.
http://arxiv.org/abs/0804.1379
---------------------------------------------------------------
6898. ORDER OF CURRENT VARIANCE AND DIFFUSIVITY IN THE RATE ONE
TOTALLY ASYMMETRIC ZERO RANGE PROCESS
Marton Balazs and Julia Komjathy
We prove that the variance of the current across a characteristic is
of order
t^{2/3} in a stationary constant rate totally asymmetric zero range
process,
and that the diffusivity has order t^{1/3}. This is a step towards
proving
universality of this scaling behavior in the class of one-dimensional
interacting systems with one conserved quantity and concave
hydrodynamic flux.
The proof proceeds via couplings to show the corresponding moment
bounds for a
second class particle. We build on the methods developed by Balazs-
Seppalainen
for asymmetric simple exclusion. However, some modifications were
needed to
handle the larger state space. Our results translate into t^{2/3}-
order of
variance of the tagged particle on the characteristics of totally
asymmetric
simple exclusion.
http://arxiv.org/abs/0804.1397
---------------------------------------------------------------
6899. A LOWER BOUND FOR THE PRINCIPAL EIGENVALUE OF THE STOKES
OPERATOR IN A RANDOM DOMAIN
V. V. Yurinsky
This article is dedicated to localization of the principal eigenvalue
(PE) of
the Stokes operator acting on solenoidal vector fields that vanish
outside a
large random domain modeling the pore space in a cubic block of porous
material
with disordered micro-structure. Its main result is an asymptotically
deterministic lower bound for the PE of the sum of a low compressibility
approximation to the Stokes operator and a small scaled random
potential term,
which is applied to produce a similar bound for the Stokes PE. The
arguments
are based on the method proposed by F. Merkl and M. V. W\"{u}trich for
localization of the PE of the Schr\"{o}dinger operator in a similar
setting.
Some additional work is needed to circumvent the complications arising
from the
restriction to divergence-free vector fields of the class of test
functions in
the variational characterization of the Stokes PE.
http://arxiv.org/abs/0804.1415
---------------------------------------------------------------
6900. AN ASYMPTOTIC RESULT FOR BROWNIAN POLYMERS
Thomas Mountford and Pierre Tarr\`es
We consider a model of the shape of a growing polymer introduced by
Durrett
and Rogers (Probab. Theory Related Fields 92 (1992) 337--349). We
prove their
conjecture about the asymptotic behavior of the underlying continuous
process
$X_t$ (corresponding to the location of the end of the polymer at time
$t$) for
a particular type of repelling interaction function without compact
support.
http://arxiv.org/abs/0804.1431
---------------------------------------------------------------
6901. QUENCHED LARGE DEVIATIONS FOR MULTIDIMENSIONAL RANDOM WALK IN
RANDOM ENVIRONMENT: A VARIATIONAL FORMULA
Jeffrey M. Rosenbluth
We take the point of view of the particle in a multidimensional nearest
neighbor random walk in random environment (RWRE). We prove a quenched
large
deviation principle and derive a variational formula for the quenched
rate
function. Most of the previous results in this area rely on the
subadditive
ergodic theorem. We employ a different technique which is based on a
minimax
theorem. Large deviation principles for RWRE have been proven for i.i.d.
nestling environments subject to a moment condition and for ergodic
uniformly
elliptic environments. We assume only that the environment is ergodic
and the
transition probabilities satisfy a moment condition.
http://arxiv.org/abs/0804.1444
---------------------------------------------------------------
6902. STABILIT\'{E} DU COMPORTEMENT DES MARCHES AL\'{E}ATOIRES SUR UN
GROUPE LOCALEMENT COMPACT
Driss Gretete
Dans cet article nous d\'{e}montrons un th\'{e}or\`{e}me de stabilit
\'{e} des
probabilit\'{e}s de retour sur un groupe localement compact
unimodulaire,
s\'{e}parable et compactement engendr\'{e}. Nous d\'{e}montrons que le
comportement asymptotique de $F^{*(2n)}(e)$ ne d\'{e}pend pas de la
densit\'{e}
$F$ sous des hypoth\`{e}ses naturelles. A titre d'exemple nous
\'{e}tablissons
que la probabilit\'{e} de retour sur une large classe de groupes r
\'{e}solubles
se comporte comme $\exp(-n^{1/3})$.
http://arxiv.org/abs/0804.1461
---------------------------------------------------------------
6903. ON ESTIMATION AND OPTIMIZATION OF PROBABILITY
Xinjia Chen
In this paper, we develop a general approach for probabilistic
estimation and
optimization. An explicit formula is derived for controlling the
reliability of
probabilistic estimation based on a mixed criterion of absolute and
relative
errors. By employing the Chernoff bound and the concept of sampling, the
minimization of a probabilistic function is transformed into an
optimization
problem amenable for gradient descendent algorithms.
http://arxiv.org/abs/0804.1399
---------------------------------------------------------------
6904. BOUNDEDNESS OF RIESZ TRANSFORMS FOR ELLIPTIC OPERATORS ON
ABSTRACT WIENER SPACES
Jan Maas and Jan van Neerven
Let (E,H,mu) be an abstract Wiener space and let D_V := VD, where D
denotes
the Malliavin derivative in the direction of H and V is a closed and
densely
defined operator from H into another Hilbert space G. Given a bounded
operator
B on G, coercive on the closure of the range of V, we consider the
realisation
of the operator D_V* B D_V in L^p(E,mu) for 1<p<\infty.
Our main result states that the following assertions are
equivalent: (1)
dom(D_V* B D_V) = dom(D_V) and Meyer's inequalities hold for D_V* B
D_V; (2)
D_V D_V* B admits a bounded H-infinity calculus on the closure of the
range of
D_V; (3) dom(V*BV) = dom(V) and Meyer's inequalities hold for V*BV;
(4) VV*B
admits a bounded H-infinity calculus on the closure of the range of V.
This result is can be viewed as a non-symmetric generalisation of the
classical Meyer inequalities (which correspond to the case G=H, V=I,
B=I). A
one-sided version of the main result, giving L^p-boundedness of the
associated
Riesz transforms in terms of a square function estimate, is also
obtained.
As an application let -A generate an analytic C_0-contraction
semigroup on a
Hilbert space H and let -L be the L^p-realisation of the generator of
its
second quantisation. Our results imply that two-sided bounds for the
Riesz
transform of L are equivalent with the Kato square root property for A.
http://arxiv.org/abs/0804.1432
---------------------------------------------------------------
6905. ON AN IDENTITY OF KY FAN
Michel Weber
We give several applications of an identity for sums of weakly
stationary
sequences due to Ky Fan.
http://arxiv.org/abs/0804.1508
---------------------------------------------------------------
6906. THE PHASE DIAGRAM OF THE QUANTUM CURIE-WEISS MODEL
Lincoln Chayes and Nicholas Crawford and Dmitry Ioffe and and Anna
Levit
This paper studies a generalization of the Curie-Weiss model (the
Ising model
on a complete graph) to quantum mechanics. Using a natural probabilistic
representation of this model, we give a complete picture of the phase
diagram
of the model in the parameters of inverse temperature and transverse
field
strength. Further analysis computes the critical exponent for the
decay of the
order parameter in the approach to the critical curve and gives useful
stability properties of a variational problem associated with the
representation.
http://arxiv.org/abs/0804.1605
---------------------------------------------------------------
6907. EXACT CONDITIONS FOR NO RUIN FOR THE GENERALISED ORNSTEIN-
UHLENBECK PROCESS
Damien Bankovsky and Allan Sly
For a bivariate L\'evy process $(\xi_t,\eta_t)_{t\geq 0}$ the
generalised
Ornstein-Uhlenbeck (GOU) process is defined as
V_t:=e^{\xi_t}(z+\int_0^t e^{-\xi_{s-}}d\eta_s), t\ge0,
where $z\in\mathbb{R}.$ We define necessary and sufficient
conditions under
which the infinite horizon ruin probability for the process is zero.
These
conditions are stated in terms of the canonical characteristics of the
L\'evy
process and reveal the effect of the dependence relationship between $
\xi$ and
$\eta.$ We also present technical results which explain the structure
of the
lower bound of the GOU.
http://arxiv.org/abs/0804.1634
---------------------------------------------------------------
6908. CALIBRATION OF TRANSPARENCY RISKS: A NOTE
Jir\^o Akahori and Yuuki Kanashi and Yuichi Morimura
The aim of this research is to give a simple framework to evaluate/
quantize
the "transparency" of a firm. We assume that the process of the firm
value is
only observable once in a while but is strongly correlated with the
stock price
which is observable and tradable. This hybrid type structure make the
transparency "observable". The implication of the present study is
that the
depth of the shock to the market caused by the precise accounting
information
does reflect the degree of transparency. Furthermore, it can be
quantized
resorting to the calibration method.
http://arxiv.org/abs/0804.1642
---------------------------------------------------------------
6909. ON PERCOLATION IN RANDOM GRAPHS WITH GIVEN VERTEX DEGREES
Svante Janson
We study the random graph obtained by random deletion of vertices or
edges
from a random graph with given vertex degrees. A simple trick of
exploding
vertices instead of deleting them, enables us to derive results from
known
results for random graphs with given vertex degrees. This is used to
study
existence of giant component and existence of k-core. As a variation
of the
latter, we study also bootstrap percolation in random regular graphs.
We obtain both simple new proofs of known results and new results. An
interesting feature is that for some degree sequences, there are
several or
even infinitely many phase transitions for the k-core.
http://arxiv.org/abs/0804.1656
---------------------------------------------------------------
6910. LOWER BOUNDS OF MARTINGALE MEASURE DENSITIES IN THE DALANG-
MORTON-WILLINGER THEOREM
Dmitry B. Rokhlin
For a $d$-dimensional stochastic process $(S_n)_{n=0}^N$ we obtain
criteria
for the existence of an equivalent martingale measure, whose density $z
$, up to
a normalizing constant, is bounded from below by a given random
variable $f$.
We consider the case of one-period model (N=1) under the assumptions $S
\in
L^p$; $f,z\in L^q$, $1/p+1/q=1$, where $p\in [1,\infty]$, and the case
of
$N$-period model for $p=\infty$. The mentioned criteria are expressed
in terms
of the conditional distributions of the increments of $S$, as well as
in terms
of the boundedness from above of an utility function related to some
optimal
investment problem under the loss constraints. Several examples are
presented.
http://arxiv.org/abs/0804.1761
---------------------------------------------------------------
6911. CROSSING PROBABILITIES IN ASYMMETRIC EXCULSION PROCESSES
Pablo A. Ferrari and Patricia Goncalves and James B. Martin
We consider the one-dimensional asymmetric simple exclusion process in
which
particles jump to the right at rate p and to the left at rate 1-p,
interacting
by exclusion. Suppose that the initial state has first-class particles
to the
left of the origin, a second-class particle at the origin, a third-class
particle at site 1 and holes to the right of site 1. We show that the
probability that the second-class particle overtakes the third-class
particle
is (1+p)/3p. We obtain various limiting results about the joint
behavior of the
second-class and third-class particles, and a partial extension to a
system
with a further (fourth-class) particle.
http://arxiv.org/abs/0804.1770
---------------------------------------------------------------
6912. SUPPRESSION OF UNBOUNDED GRADIENTS IN SDE ASSOCIATED WITH THE
BURGERS EQUATION
Sergio Albeverio and Olga Rozanova
We consider the Langevin equation describing a stochastically perturbed
non-viscous Burgers fluid and introduce a deterministic function that
corresponds to the mean of the velocity when we keep the value of
position
fixed. We study interrelations between this function and the solution
of the
non-perturbed Burgers equation. Especially we are interested in the
property of
the solution of the latter equation to develop unbounded gradients
within a
finite time. We study the question how the initial distribution of
particles
for the Langevin equation influences this blowup phenomenon. The
simplest model
case of a linear initial velocity is considered in details. We show
that if the
initial distribution of particles is uniform, then the mean of the
velocity for
a given position coincides with the solution of the Burgers equation
and in
particular does not depend on the variance of the stochastic
perturbation.
Further, for a one space space variable we get the following result:
if the
decay rate of the initial particles distribution at infinity is
greater or
equal $|x|^{-2},$ then the blowup is suppressed, otherwise, the blowup
takes
place at the same moment of time as in the case of the non-perturbed
Burgers
equation. We consider also the case of bounded initial velocity and
show, both
analytically and numerically, that for the class of initial
distribution of
particles with power-behaved decay/increase at infinity the unbounded
gradients
are eliminated.
http://arxiv.org/abs/0804.1553
---------------------------------------------------------------
6913. LAST PASSAGE PERCOLATION IN MACROSCOPICALLY INHOMOGENEOUS MEDIA
Leonardo T. Rolla and Augusto Q. Teixeira
In this note we investigate the last passage percolation model in the
presence of macroscopic inhomogeneity. We analyze how this affects the
scaling
limit of the passage time, leading to a variational problem that
provides an
ODE for the deterministic limiting shape of the maximal path. We
obtain a
sufficient analytical condition for uniqueness of the solution for the
variational problem. Consequences for the totally asymmetric simple
exclusion
process are discussed.
http://arxiv.org/abs/0804.1810
---------------------------------------------------------------
6914. INDEPENDENCE OF FOUR PROJECTIVE CRITERIA FOR THE WEAK
INVARIANCE PRINCIPLE
Olivier Durieu
Let $(X_i)_{i\in\Z}$ be a regular stationary process for a given
filtration.
The weak invariance principle holds under the condition
$\sum_{i\in\Z}\|P_0(X_i)\|_2<\infty$ (see Hannan (1979)}, Dedecker and
Merlev\`ede (2003), Deddecker, Merlev\'ede and Voln\'y (2007)). In
this paper,
we show that this criterion is independent of other known criteria: the
martingale-coboundary decomposition of Gordin (see Gordin (1969,
1973)), the
criterion of Dedecker and Rio (see Dedecker and Rio (2000)) and the
condition
of Maxwell and Woodroofe (see Maxwell and Woodroofe (2000), Peligrade
and Utev
(2005), Voln\'y (2006, 2007)).
http://arxiv.org/abs/0804.1848
---------------------------------------------------------------
6915. SMALL DEVIATIONS OF STABLE PROCESSES AND ENTROPY OF THE
ASSOCIATED RANDOM OPERATORS
Frank Aurzada and Mikhail Lifshits and Werner Linde
We investigate the relation between the small deviation problem for a
symmetric $\alpha$-stable random vector in a Banach space and the metric
entropy properties of the operator generating it. This generalizes
former
results due to Li and Linde and to Aurzada. It is shown that this
problem is
related to the study of the entropy numbers of a certain random
operator. In
some cases an interesting gap appears between the entropy of the
original
operator and that of the random operator generated by it. This
phenomenon is
studied thoroughly for diagonal operators. Basic ingredient here are
techniques
related to random partitions of the integers. The main result about
metric
entropy and small deviation allows us to determine or provide new
estimates for
the small deviations rate for several symmetric $\alpha$-stable random
processes, among them unbounded Riemann-Liouville processes, weighted
Riemann-Liouville processes, and the ($d$-dimensional) $\alpha$-stable
sheet.
http://arxiv.org/abs/0804.1883
---------------------------------------------------------------
6916. A STOCHASTIC FIXED POINT EQUATION FOR WEIGHTED MINIMA AND MAXIMA
Gerold Alsmeyer and Uwe R\"osler
Given any finite or countable collection of real numbers $T_j,j\in J$,
we
find all solutions $F$ to the stochastic fixed point equation
\[W\stackrel{\mathrm {d}}{=}\inf_{j\in J}T_jW_j,\] where $W$ and the
$W_j,j\in
J$, are independent real-valued random variables with distribution $F$
and
$\stackrel{\mathrm {d}}{=}$ means equality in distribution. The bulk
of the
necessary analysis is spent on the case when $|J|\geq 2$ and all $T_j$
are
(strictly) positive. Nontrivial solutions are then concentrated on
either the
positive or negative half line. In the most interesting (and difficult)
situation $T$ has a characteristic exponent $\alpha$ given by $\sum_{j
\in
J}T_j^{\alpha}=1$ and the set of solutions depends on the closed
multiplicative
subgroup of $\mathbb {R}^{>}=(0,\infty)$ generated by the $T_j$ which
is either
$\{1\}$, $\mathbb {R}^{>}$ itself or $r^{\mathbb {Z}}=\{r^n\dvt n\in
\mathbb
{Z}\}$ for some $r>1$. The first case being trivial, the nontrivial
fixed
points in the second case are either Weibull distributions or their
reciprocal
reflections to the negative half line (when represented by random
variables),
while in the third case further periodic solutions arise. Our analysis
builds
on the observation that the logarithmic survival function of any fixed
point is
harmonic with respect to $\varLambda =\sum_{j\geq 1}\delta_{T_j}$, i.e.
$\varGamma =\varGamma \star \varLambda$, where $\star$ means
multiplicative
convolution. This will enable us to apply the powerful Choquet--Deny
theorem.
http://arxiv.org/abs/0804.1884
---------------------------------------------------------------
6917. MULTIVARIATE NORMAL APPROXIMATION USING STEIN'S METHOD AND
MALLIAVIN CALCULUS
Ivan Nourdin (PMA) and Giovanni Peccati (LSTA) and Anthony R
\'eveillac (LMA-Rochelle)
We combine Stein's method with Malliavin calculus in order to obtain
explicit
bounds in the multidimensional normal approximation (in the Wasserstein
distance) of functionals of Gaussian fields. Our results generalize
and refine
the main findings by Peccati and Tudor (2005), Nualart and Ortiz-Latorre
(2007), Peccati (2007) and Nourdin and Peccati (2007b, 2008); in
particular,
they apply to approximations by means of Gaussian vectors with an
arbitrary,
positive definite covariance matrix. Among several examples, we
provide an
application to a functional version of the Breuer-Major CLT for fields
subordinated to a fractional Brownian motion.
http://arxiv.org/abs/0804.1889
---------------------------------------------------------------
6918. ON PROBABILISTIC PARAMETRIC INFERENCE
Tomaz Podobnik and Tomi Zivko
An objective operational theory of probabilistic parametric inference is
formulated without invoking the so-called non-informative prior
probability
distributions.
http://arxiv.org/abs/0804.1905
---------------------------------------------------------------
6919. MEASURE AND INTEGRAL WITH PURELY ORDINAL SCALES
Dieter Denneberg and Michel Grabisch (LIP6)
We develop a purely ordinal model for aggregation functionals for
lattice
valued functions, comprising as special cases quantiles, the Ky Fan
metric and
the Sugeno integral. For modeling findings of psychological
experiments like
the reflection effect in decision behaviour under risk or uncertainty,
we
introduce reflection lattices. These are complete linear lattices
endowed with
an order reversing bijection like the reflection at 0 on the real
interval
$[-1,1]$. Mathematically we investigate the lattice of non-void
intervals in a
complete linear lattice, then the class of monotone interval-valued
functions
and their inner product.
http://arxiv.org/abs/0804.1758
---------------------------------------------------------------
6920. THE SYMMETRIC SUGENO INTEGRAL
Michel Grabisch (LIP6)
We propose an extension of the Sugeno integral for negative numbers,
in the
spirit of the symmetric extension of Choquet integral, also called
\Sipos\
integral. Our framework is purely ordinal, since the Sugeno integral
has its
interest when the underlying structure is ordinal. We begin by defining
negative numbers on a linearly ordered set, and we endow this new
structure
with a suitable algebra, very close to the ring of real numbers. In a
second
step, we introduce the M\"obius transform on this new structure.
Lastly, we
define the symmetric Sugeno integral, and show its similarity with the
symmetric Choquet integral.
http://arxiv.org/abs/0804.1760
---------------------------------------------------------------
6921. STATIONARY DISTRIBUTIONS FOR DIFFUSIONS WITH INERT DRIFT
Richard F. Bass and Krzysztof Burdzy and Zhen-Qing Chen and Martin
Hairer
Consider a reflecting diffusion in a domain in $R^d$ that acquires
drift in
proportion to the amount of local time spent on the boundary of the
domain. We
show that the stationary distribution for the joint law of the
position of the
reflecting process and the value of the drift vector has a product form.
Moreover, the first component is the symmetrizing measure on the
domain for the
reflecting diffusion without inert drift, and the second component has a
Gaussian distribution. We also consider processes where the drift is
given in
terms of the gradient of a potential.
http://arxiv.org/abs/0804.2029
---------------------------------------------------------------
6922. CRITICAL BEHAVIOR AND THE LIMIT DISTRIBUTION FOR LONG-RANGE
ORIENTED PERCOLATION. II: SPATIAL CORRELATION
Lung-Chi Chen and Akira Sakai
We prove that the Fourier transform of the properly-scaled normalized
two-point function for sufficiently spread-out long-range oriented
percolation
with index \alpha>0 converges to e^{-C|k|^{\alpha\wedge2}} for some
positive
finite constant C above the upper-critical dimension 2min{\alpha,2}.
This
answers the open question remained in the previous paper (Chen and
Sakai 2008).
Moreover, we show that the constant C exhibits a crossover phenomenon at
\alpha=2. The proof is based on a new method of estimating fractional
moments
for the spatial variable of the lace-expansion coefficients.
http://arxiv.org/abs/0804.2039
---------------------------------------------------------------
6923. STOCHASTIC CHAINS WITH MEMORY OF VARIABLE LENGTH
Antonio Galves and Eva L\"ocherbach
Stochastic chains with memory of variable length constitute an
interesting
family of stochastic chains of infinite order on a finite alphabet.
The idea is
that for each past, only a finite suffix of the past, called context,
is enough
to predict the next symbol. These models were first introduced in the
information theory literature by Rissanen (1983) as a universal tool
to perform
data compression. Recently, they have been used to model up scientific
data in
areas as different as biology, linguistics and music. This paper
presents a
personal introductory guide to this class of models focusing on the
algorithm
Context and its rate of convergence.
http://arxiv.org/abs/0804.2050
---------------------------------------------------------------
6924. SURFACE TENSION IN THE DILUTE ISING MODEL. THE WULFF CONSTRUCTION
Marc Wouts (MODAL'x)
We study the surface tension and the phenomenon of phase coexistence
for the
Ising model on $\mathbbm{Z}^d$ ($d \geqslant 2$) with ferromagnetic
but random
couplings. We prove the convergence in probability (with respect to
random
couplings) of surface tension and analyze its large deviations : upper
deviations occur at volume order while lower deviations occur at
surface order.
We study the asymptotics of surface tension at low temperatures and
relate the
quenched value $\tau^q$ of surface tension to maximal flows (first
passage
times if $d = 2$). For a broad class of distributions of the couplings
we show
that the inequality $\tau^a \leqslant \tau^q$ -- where $\tau^a$ is the
surface
tension under the averaged Gibbs measure -- is strict at low
temperatures. We
also describe the phenomenon of phase coexistence in the dilute Ising
model and
discuss some of the consequences of the media randomness. All of our
results
hold as well for the dilute Potts and random cluster models.
http://arxiv.org/abs/0804.2208
---------------------------------------------------------------
6925. STONE-WEIERSTRASS TYPE THEOREMS FOR LARGE DEVIATIONS
Henri Comman
We give a general version of Bryc's theorem valid on any topological
space
and with any algebra $\mathcal{A}$ of real-valued continuous functions
separating the points, or any well-separating class. In absence of
exponential
tightness, and when the underlying space is locally compact regular and
$\mathcal{A}$ constituted by functions vanishing at infinity, we give a
sufficient condition on the functional $\Lambda(\cdot)_{\mid
\mathcal{A}}$ to
get large deviations with not necessarily tight rate function. We
obtain the
general variational form of any rate function on a completely regular
space;
when either exponential tightness holds or the space is locally compact
Hausdorff, we get it in terms of any algebra as above. Prohorov-type
theorems
are generalized to any space, and when it is locally compact regular the
exponential tightness can be replaced by a (strictly weaker) condition
on
$\Lambda(\cdot)_{\mid \mathcal{A}}$.
http://arxiv.org/abs/0804.2214
---------------------------------------------------------------
6926. TWO-DIMENSIONAL MARKOVIAN HOLONOMY FIELDS
Thierry L\'evy (DMA)
We define a notion of Markov process indexed by curves drawn on a
compact
surface and taking its values in a compact Lie group. We call such a
process a
two-dimensional Markovian holonomy field. The prototype of this class of
processes, and the only one to have been constructed before the
present work,
is the canonical process under the Yang-Mills measure, first defined
by Ambar
Sengupta and later by the author . The Yang-Mills measure sits in the
class of
Markovian holonomy fields very much like the Brownian motion in the
class of
Levy processes. We prove that every regular Markovian holonomy field
determines
a Levy process of a certain class on the Lie group in which it takes its
values, and construct, for each Levy process in this class, a Markovian
holonomy field to which it is associated. When the Lie group is in
fact a
finite group, we give an alternative construction of this Markovian
holonomy
field as the monodromy of a random ramified principal bundle.
http://arxiv.org/abs/0804.2230
---------------------------------------------------------------
6927. A CONSTRUCTIVE PROOF OF THE EXISTENCE OF VITERBI PROCESSES
J. Lember and A. Koloydenko
Since the early days of digital communication, hidden Markov models
(HMMs)
have now been also routinely used in speech recognition, processing of
natural
languages, images, and in bioinformatics. In an HMM $(X_i,Y_i)_{i\ge
1}$,
observations $X_1,X_2,...$ are assumed to be conditionally independent
given an
``explanatory'' Markov process $Y_1,Y_2,...$, which itself is not
observed;
moreover, the conditional distribution of $X_i$ depends solely on $Y_i$.
Central to the theory and applications of HMM is the Viterbi algorithm
to find
{\em a maximum a posteriori} (MAP) estimate $q_{1:n}=(q_1,q_2,...,q_n)
$ of
$Y_{1:n}$ given observed data $x_{1:n}$. Maximum {\em a posteriori}
paths are
also known as Viterbi paths or alignments. Recently, attempts have
been made to
study the behavior of Viterbi alignments when $n\to \infty$. Thus, it
has been
shown that in some special cases a well-defined limiting Viterbi
alignment
exists. While innovative, these attempts have relied on rather strong
assumptions and involved proofs which are existential. This work
proves the
existence of infinite Viterbi alignments in a more constructive manner
and for
a very general class of HMMs.
http://arxiv.org/abs/0804.2138
---------------------------------------------------------------
6928. LARGE DEVIATIONS FOR QUANTUM MARKOV SEMIGROUPS ON THE 2 X 2
MATRIX ALGEBRA
Henri Comman
Let $({\mathcal{T}}_{*t})$ be a predual quantum Markov semigroup
acting on
the full 2 x 2 matrix algebra and having an absorbing pure state. We
prove that
for any initial state $\omega$, the net of orthogonal measures
representing the
net of states $({\mathcal{T}}_{*t}(\omega))$ satisfies a large deviation
principle in the pure state space, with a rate function given in terms
of the
generator, and which does not depend on $\omega$. This implies that
$({\mathcal{T}}_{*t}(\omega))$ is faithful for all $t$ large enough.
Examples
arising in weak coupling limit are studied.
http://arxiv.org/abs/0804.2093
---------------------------------------------------------------
6929. THE SECRECY GRAPH AND SOME OF ITS PROPERTIES
Martin Haenggi
A new random geometric graph model, the so-called secrecy graph, is
introduced and studied. The graph represents a wireless network and
includes
only edges over which secure communication in the presence of
eavesdroppers is
possible. The underlying point process models considered are lattices
and
Poisson point processes. In the lattice case, analogies to standard
bond and
site percolation can be exploited to determine percolation thresholds.
In the
Poisson case, the node degrees are determined and percolation is
studied using
analytical bounds and simulations. It turns out that a small density of
eavesdroppers already has a drastic impact on the connectivity of the
secrecy
graph.
http://arxiv.org/abs/0804.2249
---------------------------------------------------------------
6930. BOUNDS FOR THE LOSS PROBABILITIES OF LARGE LOSS QUEUEING SYSTEMS
Vyacheslav M. Abramov
The aim of this paper is to establish the bounds for the least root of
the
functional equation $x=\hat{G}(\mu-\mu x)$, where $\hat{G}(s)$ is the
Laplace-Stieltjes transform of an unknown probability distribution
function
$G(x)$ of a positive random variable having the first two moments $
\frak{g}_1$
and $\frak{g}_2$, and $\mu$ is a positive parameter satisfying the
condition
$\mu\frak{g}_1>1$. The additional information characterizing $G(x)$ is
an
empirical probability distribution function ${G}_{\mathrm{emp}}(x)$,
and it is
assumed that the distance in the uniform (Kolmogorov) metric between
$G(x)$ and
${G}_{\mathrm{emp}}(x)$ is not greater than $\kappa$. The obtained
bounds for
the positive least root of the functional equation $x=\hat{G}(\mu-\mu
x)$ are
then used to find the asymptotic bounds for the loss probabilities in
certain
queueing systems with a large number of waiting places, when only an
empirical
probability distribution function of an interarrival or service time
is known.
http://arxiv.org/abs/0804.2310
---------------------------------------------------------------
6931. SPECIAL INVITED PAPER. LARGE DEVIATIONS
S. R. S. Varadhan
This paper is based on Wald Lectures given at the annual meeting of
the IMS
in Minneapolis during August 2005. It is a survey of the theory of large
deviations.
http://arxiv.org/abs/0804.2330
---------------------------------------------------------------
6932. ALMOST SURE EULER HYDRODYNAMICS OF ONE-DIMENSIONAL ATTRACTIVE
PARTICLE SYSTEMS
C. Bahadoran (1) and H. Guiol (2) and K. Ravishankar (3) and E.
Saada (4) ((1) Univ. Clermont-Ferrand France, (2) Grenoble Univ.
France, (3) SUNY USA,
(4) CNRS-Rouen France)
We consider attractive irreducible conservative particle systems on Z,
with
at most K particles per site, for which no explicit invariant measures
are
required. We suppose that jumps have a finite positive first moment. In
Bahadoran et al. (2006) we proved, under finite range hypothesis, that
for such
systems the hydrodynamic limit under Euler scaling exists, and is
given by the
entropy solution of a scalar conservation law with Lipschitz-
continuous flux.
Here, by a refinement of our method, we obtain an almost sure
hydrodynamic
limit, when starting from: i) any shock profile (Riemann
hydrodynamics); ii)
any general initial profile, but with finite range assumption.
http://arxiv.org/abs/0804.2345
---------------------------------------------------------------
6933. ALMOST-SURE GROWTH RATE OF GENERALIZED RANDOM FIBONACCI SEQUENCES
Elise Janvresse (LMRS) and Beno\^it Rittaud (IG and LMPT) and
Thierry De La Rue (LMRS)
We study the generalized random Fibonacci sequences defined by their
first
nonnegative terms and for $n\ge 1$, $F_{n+2} = \lambda F_{n+1} \pm
F_{n}$
(linear case) and $\widetilde F_{n+2} = |\lambda \widetilde F_{n+1} \pm
\widetilde F_{n}|$ (non-linear case), where each $\pm$ sign is
independent and
either $+$ with probability $p$ or $-$ with probability $1-p$ ($0<p\le
1$). Our
main result is that, when $\lambda$ is of the form $\lambda_k = 2\cos
(\pi/k)$
for some integer $k\ge 3$, the exponential growth of $F_n$ for $0<p\le
1$, and
of $\widetilde F_{n}$ for $1/k < p\le 1$, is almost surely positive
and given
by $$ \int_0^\infty \log x d\nu_{k, \rho} (x), $$ where $\rho$ is an
explicit
function of $p$ depending on the case we consider, taking values in
$[0, 1]$,
and $\nu_{k, \rho}$ is an explicit probability distribution on $\RR_+$
defined
inductively on generalized Stern-Brocot intervals. We also provide an
integral
formula for $0<p\le 1$ in the easier case $\lambda\ge 2$. Finally, we
study the
variations of the exponent as a function of $p$.
http://arxiv.org/abs/0804.2378
---------------------------------------------------------------
6934. GROWTH RATE FOR THE EXPECTED VALUE OF A GENERALIZED RANDOM
FIBONACCI SEQUENCE
Elise Janvresse (LMRS) and Beno\^it Rittaud (IG and LMPT) and
Thierry De La Rue (LMRS)
A random Fibonacci sequence is defined by the relation g_n = | g_{n-1}
+/-
g_{n-2} |, where the +/- sign is chosen by tossing a balanced coin for
each n.
We generalize these sequences to the case when the coin is unbalanced
(denoting
by p the probability of a +), and the recurrence relation is of the
form g_n =
|\lambda g_{n-1} +/- g_{n-2} |. When \lambda >=2 and 0 < p <= 1, we
prove that
the expected value of g_n grows exponentially fast. When \lambda =
\lambda_k =
2 cos(\pi/k) for some fixed integer k>2, we show that the expected
value of g_n
grows exponentially fast for p>(2-\lambda_k)/4 and give an algebraic
expression
for the growth rate. The involved methods extend (and correct) those
introduced
in a previous paper by the second author.
http://arxiv.org/abs/0804.2400
---------------------------------------------------------------
6935. DYNAMICAL LARGE DEVIATIONS FOR THE BOUNDARY DRIVEN WEAKLY
ASYMMETRIC EXCLUSION PROCESS
Lorenzo Bertini and Claudio Landim and Mustapha Mourragui
We consider the weakly asymmetric exclusion process on a bounded
interval
with particles reservoirs at the endpoints. The hydrodynamic limit for
the
empirical density, obtained in the diffusive scaling, is given by the
viscous
Burgers equation with Dirichlet boundary conditions. We prove the
associated
dynamical large deviations principle.
http://arxiv.org/abs/0804.2458
---------------------------------------------------------------
6936. ON THE PERMANENT OF RANDOM BERNOULLI MATRICES
T. Tao and V. Vu
We show that the permanent of an $n \times n$ matrix with iid Bernoulli
entries $\pm 1$ is of magnitude $n^{({1/2}+o(1))n}$ with probability
$1-o(1)$.
In particular, it is almost surely non-zero.
http://arxiv.org/abs/0804.2362
---------------------------------------------------------------
6937. ERROR BOUNDS ON THE NON-NORMAL APPROXIMATION OF HERMITE POWER
VARIATIONS OF FRACTIONAL BROWNIAN MOTION
Jean-Christophe Breton (LMA-Rochelle) and Ivan Nourdin (PMA)
Let $q\geq 2$ be a positive integer, $B$ be a fractional Brownian
motion with
Hurst index $H\in(0,1)$, $Z$ be an Hermite random variable of index $q
$, and
$H_q$ denote the Hermite polynomial having degree $q$. For any $n\geq
1$, set
$V_n=\sum_{k=0}^{n-1} H_q(B_{k+1}-B_k)$. The aim of the current paper
is to
derive, in the case when the Hurst index verifies $H>1-1/(2q)$, an
upper bound
for the total variation distance between the laws $\mathscr{L}(Z_n)$ and
$\mathscr{L}(Z)$, where $Z_n$ stands for the correct renormalization
of $V_n$
which converges in distribution towards $Z$. Our results should be
compared
with those obtained recently by Nourdin and Peccati (2007) in the case
when
$H<1-1/(2q)$, corresponding to the situation where one has normal
approximation.
http://arxiv.org/abs/0804.2528
---------------------------------------------------------------
6938. MAX-PLUS DECOMPOSITION OF SUPERMARTINGALES AND CONVEX ORDER.
APPLICATION TO AMERICAN OPTIONS AND PORTFOLIO INSURANCE
Nicole El Karoui and Asma Meziou
We are concerned with a new type of supermartingale decomposition in the
Max-Plus algebra, which essentially consists in expressing any
supermartingale
of class $(\mathcal{D})$ as a conditional expectation of some running
supremum
process. As an application, we show how the Max-Plus supermartingale
decomposition allows, in particular, to solve the American optimal
stopping
problem without having to compute the option price. Some illustrative
examples
based on one-dimensional diffusion processes are then provided. Another
interesting application concerns the portfolio insurance. Hence, based
on the
``Max-Plus martingale,'' we solve in the paper an optimization problem
whose
aim is to find the best martingale dominating a given floor process
(on every
intermediate date), w.r.t. the convex order on terminal values.
http://arxiv.org/abs/0804.2561
---------------------------------------------------------------
6939. THE ALLELIC PARTITION FOR COALESCENT POINT PROCESSES
Amaury Lambert (CMAP and Fese)
Assume that individuals alive at time $t$ in some population can be
ranked in
such a way that the coalescence times between consecutive individuals
are
i.i.d. The ranked sequence of these branches is called a coalescent
point
process. We have shown in a previous work that splitting trees are
important
instances of such populations. Here, individuals are given DNA
sequences, and
for a sample of $n$ DNA sequences belonging to distinct individuals, we
consider the number $S_n$ of polymorphic sites (sites at which at
least two
sequences differ), and the number $A_n$ of distinct haplotypes
(sequences
differing at one site at least). It is standard to assume that
mutations arrive
at constant rate (on germ lines), and never hit the same site on the DNA
sequence. We study the mutation pattern associated to coalescent point
processes under this assumption. Here, $S_n$ and $A_n$ grow linearly
as $n$
grows, with explicit rate. However, when the branch lengths have
infinite
expectation, $S_n$ grows more rapidly, e.g. as $n \ln(n)$ for critical
birth--death processes. Then, we study the frequency spectrum of the
sample,
that is, the numbers of polymorphic sites/haplotypes carried by $k$
individuals
in the sample. These numbers are shown to grow also linearly with
sample size,
and we provide simple explicit formulae for mutation frequencies and
haplotype
frequencies. For critical birth--death processes, mutation frequencies
are
given by the harmonic series and haplotype frequencies by Fisher
logarithmic
series.
http://arxiv.org/abs/0804.2572
---------------------------------------------------------------
6940. ON THE ORTHOGONAL POLYNOMIALS ASSOCIATED WITH A L\'EVY PROCESS
Josep Llu\'is Sol\'e and Frederic Utzet
Let $X=\{X_t, t\ge0\}$ be a c\`{a}dl\`{a}g L\'{e}vy process, centered,
with
moments of all orders. There are two families of orthogonal polynomials
associated with $X$. On one hand, the Kailath--Segall formula gives the
relationship between the iterated integrals and the variations of
order $n$ of
$X$, and defines a family of polynomials $P_1(x_1), P_2(x_1,x_2),...$
that are
orthogonal with respect to the joint law of the variations of $X$. On
the other
hand, we can construct a sequence of orthogonal polynomials
$p^{\sigma}_n(x)$
with respect to the measure $\sigma^2\delta_0(dx)+x^2 \nu(dx)$, where
$\sigma^2$ is the variance of the Gaussian part of $X$ and $\nu$ its L
\'{e}vy
measure. These polynomials are the building blocks of a kind of chaotic
representation of the square functionals of the L\'{e}vy process
proved by
Nualart and Schoutens. The main objective of this work is to study the
probabilistic properties and the relationship of the two families of
polynomials. In particular, the L\'{e}vy processes such that the
associated
polynomials $P_n(x_1,...,x_n)$ depend on a fixed number of variables are
characterized. Also, we give a sequence of L\'{e}vy processes that
converge in
the Skorohod topology to $X$, such that all variations and iterated
integrals
of the sequence converge to the variations and iterated integrals of $X
$.
http://arxiv.org/abs/0804.2585
---------------------------------------------------------------
6941. NONSTANDARD LIMIT THEOREM FOR INFINITE VARIANCE FUNCTIONALS
Allan Sly and Chris Heyde
We consider functionals of long-range dependent Gaussian sequences with
infinite variance and obtain nonstandard limit theorems. When the long-
range
dependence is strong enough, the limit is a Hermite process, while for
weaker
long-range dependence, the limit is $\alpha$-stable L\'{e}vy motion.
For the
critical value of the long-range dependence parameter, the limit is a
sum of a
Hermite process and $\alpha$-stable L\'{e}vy motion.
http://arxiv.org/abs/0804.2588
---------------------------------------------------------------
6942. SHAPE TRANSITION UNDER EXCESS SELF-INTERSECTIONS FOR TRANSIENT
RANDOM WALK
Amine Asselah
We reveal a phenomenon of transition in the geometry of a transient
simple
random walk forced to realize an excess q-fold self-intersection, as the
strength parameter, q, is continuously increased. Also, as an
application of
our approach, we establish a central limit theorem for the q-fold
self-intersection in dimension 4 ore more.
http://arxiv.org/abs/0804.2616
---------------------------------------------------------------
6943. ON THE PROBABILISTIC DESCRIPTION OF A MULTIPARTITE CORRELATION
EXPERIMENT WITH ARBITRARY NUMBERS OF SETTINGS AND OUTCOMES PER SITE
Elena R. Loubenets
We consistently formalize the probabilistic description of
multipartite joint
measurements performed on systems of any nature. This allows us: (1)
to specify
in probabilistic terms the difference between nonsignaling, the
Einstein-
Podolsky-Rosen (EPR) locality and Bell's locality; (2) to introduce
the notion
of an LHV model for an S_{1}x...xS_{N}-setting N-partite correlation
experiment, with outcomes of any spectral type, discrete or
continuous, and to
prove both general and specific "quantum" statements on an LHV
simulation in an
arbitrary multipartite case; (3) to classify LHV models for a
multipartite
quantum state, in particular, to show that any N-partite quantum
state, pure or
mixed, admits an Sx1x...x1 -setting LHV description; (4) to evaluate a
threshold visibility for a noisy bipartite quantum state to admit an
S_{1}xS_
{2}-setting LHV description under any generalized quantum measurements
of two
parties. In a sequel to this paper, we shall introduce a single general
representation incorporating in a unique manner all Bell-type
inequalities for
either joint probabilities or correlation functions that have been
introduced
or will be introduced in the literature.
http://arxiv.org/abs/0804.2398
---------------------------------------------------------------
6944. THE ERGODIC DECOMPOSITION OF ASYMPTOTICALLY MEAN STATIONARY
RANDOM SOURCES
Alexander Schoenhuth
It is demonstrated how to represent asymptotically mean stationary (AMS)
random sources with values in standard spaces as mixtures of ergodic AMS
sources. This an extension of the well known decomposition of stationary
sources which has facilitated the generalization of prominent source
coding
theorems to arbitrary, not necessarily ergodic, stationary sources.
Asymptotic
mean stationarity generalizes the definition of stationarity and
covers a much
larger variety of real-world examples of random sources of practical
interest.
It is sketched how to obtain source coding and related theorems for
arbitrary,
not necessarily ergodic, AMS sources, based on the presented ergodic
decomposition.
http://arxiv.org/abs/0804.2487
---------------------------------------------------------------
6945. POISSON PROCESSES FOR SUSBSYSTEMS OF FINITE TYPE IN SYMBOLIC
DYNAMICS
J.-R. Chazottes and Z. Coelho and P. Collet
Let $\Delta\subsetneq\V$ be a proper subset of the vertices $\V$ of the
defining graph of an irreducible and aperiodic shift of finite type
$(\Sigma_{A}^{+},\S)$. Let $\Sigma_{\Delta}$ be the subshift of
allowable paths
in the graph of $\Sigma_{A}^{+}$ which only passes through the
vertices of
$\Delta$. For a random point $x$ chosen with respect to an equilibrium
state
$\mu$ of a H\"older potential $\phi$ on $\Sigma_{A}^{+}$, let $\tau_{n}
$ be the
point process defined as the sum of Dirac point masses at the times
$k>0$,
suitably rescaled, for which the first $n$-symbols of $\S^k x$ belong to
$\Delta$. We prove that this point process converges in law to a
marked Poisson
point process of constant parameter measure. The scale is related to the
pressure of the restriction of $\phi$ to $\Sigma_{\Delta}$ and the
parameters
of the limit law are explicitly computed.
http://arxiv.org/abs/0804.2550
---------------------------------------------------------------
6946. ON THE ASYMPTOTIC MEASURE OF PERIODIC SUBSYSTEMS OF FINITE TYPE
IN SYMBOLIC DYNAMICS
J.-R. Chazottes and Z. Coelho and P. Collet
Let $\Delta\subsetneq\V$ be a proper subset of the vertices $\V$ of the
defining graph of an aperiodic shift of finite type $(\Sigma_{A}^{+},
\S)$. Let
$\Delta_{n}$ be the union of cylinders in $\Sigma_{A}^{+}$
corresponding to the
points $x$ for which the first $n$-symbols of $x$ belong to $\Delta$
and let
$\mu$ be an equilibrium state of a H\"older potential $\phi$ on
$\Sigma_{A}^{+}$. We know that $\mu(\Delta_{n})$ converges to zero as
$n$
diverges. We study the asymptotic behaviour of $\mu(\Delta_{n})$ and
compare it
with the pressure of the restriction of $\phi$ to $\Sigma_{\Delta}$. The
present paper extends some results in \cite{CCC} to the case when
$\Sigma_{\Delta}$ is irreducible and periodic. We show an explicit
example
where the asymptotic behaviour differs from the aperiodic case.
http://arxiv.org/abs/0804.2551
---------------------------------------------------------------
6947. CHARACTERIZATION OF COMPACT SUBSETS OF $\MATHCAL{A}^P$ WITH
RESPECT TO WEAK TOPOLOGY
Hirbod Assa
In this brief article we characterize the relatively compact subsets of
$\mathcal{A}^p$ for the topology $\sigma(\mathcal{A}^p,\mathcal{R}^q)$
(see
below), by the weak compact subsets of $L^p$ . The spaces $
\mathcal{R}^q$
endowed with the weak topology induced by $\mathcal{A}^p$, was recently
employed to create the convex risk theory of random processes. The
weak compact
sets of $\mathcal{A}^p$ are important to characterize the so-called
Lebesgue
property of convex risk measures, to give a complete description of
the Makcey
topology on $\mathcal{R}^q$ and for their use in the optimization
theory.
http://arxiv.org/abs/0804.2873
---------------------------------------------------------------
6948. UNIFORM OBSERVABILITY OF HIDDEN MARKOV MODELS AND FILTER
STABILITY FOR UNSTABLE SIGNALS
Ramon van Handel
A hidden Markov model is called observable if distinct initial laws
give rise
to distinct laws of the observation process. Observability implies
stability of
the nonlinear filter when the signal process is tight, but this need
not be the
case when the signal process is unstable. This paper introduces a
stronger
notion of uniform observability which guarantees stability of the
nonlinear
filter in the absence of stability assumptions on the signal. By
developing
certain uniform approximation properties of convolution operators, we
subsequently demonstrate that the uniform observability condition is
satisfied
for various classes of filtering models with white noise type
observations.
This includes the case of observable linear Gaussian filtering models,
so that
standard results on stability of the Kalman filter are obtained as a
special
case.
http://arxiv.org/abs/0804.2885
---------------------------------------------------------------
6949. THE CONTINUOUS BEHAVIOR OF THE NUMERAIRE PORTFOLIO UNDER SMALL
CHANGES IN INFORMATION STRUCTURE, PROBABILISTIC VIEWS AND INVESTMENT
CONSTRAINTS
Constantinos Kardaras
The numeraire portfolio in a financial market is the unique positive
wealth
process that makes all other nonnegative wealth processes
supermartingales,
when deflated by it. The numeraire portfolio depends on market
characteristics,
which include: (a) the information flow available to acting agents,
given by a
filtration; (b) the statistical evolution of the asset prices and, more
generally, the states of nature, given by a probability measure; and (c)
possible restrictions that acting agents might be facing on available
investment strategies, modeled by a constraints set. In a financial
market with
continuous-path asset prices, the stable behavior of the numeraire
portfolio is
established when each of the aforementioned market parameters is
changed in an
infinitesimal way.
http://arxiv.org/abs/0804.2912
---------------------------------------------------------------
6950. GEODESICS IN LARGE PLANAR MAPS AND IN THE BROWNIAN MAP
Jean-Francois Le Gall
We study geodesics in the random metric space called the Brownian map,
which
appears as the scaling limit of large planar maps. In particular, we
completely
describe geodesics starting from the distinguished point called the
root, and
we characterize the set of all points that are connected to the root
by more
than one geodesic. We also prove that points of the Brownian map can be
connected to the root by at most three distinct geodesics. Our results
have
applications to the behavior of geodesics in large planar maps.
http://arxiv.org/abs/0804.3012
---------------------------------------------------------------
6951. ABSOLUTE CONTINUITY FOR SOME ONE-DIMENSIONAL PROCESSES
Nicolas Fournier and Jacques Printems
We introduce an elementary method for proving the absolute continuity
of the
time marginals of one-dimensional processes. It is based on a comparison
between the Fourier transform of such time marginals with those of the
one-step
Euler approximation of the underlying process. We obtain some absolute
continuity results for stochastic differential equations with H\"older
continuous coefficients. Furthermore, we allow such coefficients to be
random
and to depend on the whole path of the solution. We also show how it
can be
extended to some stochastic partial differential equations, and to some
L\'evy-driven stochastic differential equations. In the cases under
study, the
Malliavin calculus cannot be used, because the solution in generally not
Malliavin-differentiable.
http://arxiv.org/abs/0804.3037
---------------------------------------------------------------
6952. SMALL PARTS IN THE BERNOULLI SIEVE
Alexander Gnedin and Alex Iksanov and Uwe Roesler
Sampling from a random discrete distribution induced by a `stick-
breaking'
process is considered. Under a moment condition, it is shown that the
asymptotics of the sequence of occupancy numbers, and of the small-
parts counts
(singletons, doubletons, etc) can be read off from a limiting model
involving a
unit Poisson point process and a self-similar renewal process on the
halfline.
http://arxiv.org/abs/0804.3052
---------------------------------------------------------------
6953. REFINED CONVERGENCE FOR THE BOOLEAN MODEL
Pierre Calka (MAP5) and Julien Michel (UMPA-ENSL) and Katy Paroux
(LM-Besan\c{c}on, IRISA)
In a previous work, two of the authors proposed a new proof of a well
known
convergence result for the scaled elementary connected vacant
component in the
high intensity Boolean model towards the Crofton cell of the Poisson
hyperplane
process. In this paper, we consider the particular case of the two-
dimensional
Boolean model where the grains are discs with random radii. We
investigate the
second-order term in this convergence when the Boolean model and the
Poisson
line process are coupled on the same probability space. A precise
coupling
between the Boolean model and the Poisson line process is first
established, a
result of directional convergence in distribution for the difference
of the two
sets involved is derived as well.
http://arxiv.org/abs/0804.3088
---------------------------------------------------------------
6954. A CHARACTERIZATION OF DIMENSION FREE CONCENTRATION IN TERMS OF
TRANSPORTATION INEQUALITIES
Nathael Gozlan (LAMA)
The aim of this paper is to show that a probability measure concentrates
independently of the dimension like a gaussian measure if and only if it
verifies Talagrand's $\T_2$ transportation-cost inequality. This theorem
permits us to give a new and very short proof of a result of Otto and
Villani.
Generalizations to other types of concentration are also considered. In
particular, one shows that the Poincar\'e inequality is equivalent to
a certain
form of dimension free exponential concentration. The proofs of these
results
rely on simple Large Deviations techniques.
http://arxiv.org/abs/0804.3089
---------------------------------------------------------------
6955. A SIMPLE SAMPLE SIZE FORMULA FOR ESTIMATING MEANS OF POISSON
RANDOM VARIABLES
Xinjia Chen
In this paper, we derive an explicit sample size formula based a mixed
criterion of absolute and relative errors for estimating means of
Poisson
random variables.
http://arxiv.org/abs/0804.3033
---------------------------------------------------------------
6956. ANCESTRAL PROCESS AND DIFFUSION MODEL WITH SELECTION
Shuhei Mano
The ancestral selection graph in population genetics introduced by
Krone and
Neuhauser (1997) is an analogue to the coalescent genealogy. The
number of
ancestral particles, backward in time, of a sample of genes is an
ancestral
process, which is a birth and death process with quadratic death and
linear
birth rate. In this paper an explicit form of the number of ancestral
particle
is obtained, by using the density of the allele frequency in the
corresponding
diffusion model obtained by Kimura (1955). It is shown that fixation is
convergence of the ancestral process to the stationary measure. The
time to
fixation of an allele is studied in terms of the ancestral process.
http://arxiv.org/abs/0804.2696
---------------------------------------------------------------
6957. BIPOLARIZATION OF POSETS AND NATURAL INTERPOLATION
Michel Grabisch (CES) and Christophe Labreuche (TRT)
The Choquet integral w.r.t. a capacity can be seen in the finite case
as a
parsimonious linear interpolator between vertices of $[0,1]^n$. We
take this
basic fact as a starting point to define the Choquet integral in a
very general
way, using the geometric realization of lattices and their natural
triangulation, as in the work of Koshevoy. A second aim of the paper
is to
define a general mechanism for the bipolarization of ordered
structures. Bisets
(or signed sets), as well as bisubmodular functions, bicapacities,
bicooperative games, as well as the Choquet integral defined for them
can be
seen as particular instances of this scheme. Lastly, an application to
multicriteria aggregation with multiple reference levels illustrates
all the
results presented in the paper.
http://arxiv.org/abs/0804.2819
---------------------------------------------------------------
6958. HITTING TIME STATISTICS AND EXTREME VALUE THEORY
Ana Cristina Moreira Freitas and Jorge Milhazes Freitas and Mike Todd
We consider discrete time dynamical system and show the link between
Hitting
Time Statistics (the distribution of the first time points land in
asymptotically small sets) and Extreme Value Theory (distribution
properties of
the partial maximum of stochastic processes). This relation allows to
study
Hitting Time Statistics with tools from Extreme Value Theory, and vice
versa.
We apply these results to non-uniformly hyperbolic systems and prove
that a
multimodal map with an absolutely continuous invariant measure must
satisfy the
classical extreme value laws (with no extra condition on the speed of
mixing,
for example). We extend these ideas to the subsequent returns to the
asymptotically small sets, linking the Poisson statistics of both
processes.
http://arxiv.org/abs/0804.2887
---------------------------------------------------------------
6959. SHUFFLING ALGORITHM FOR BOXED PLANE PARTITIONS
Alexei Borodin and Vadim Gorin
We introduce discrete time Markov chains that preserve uniform
measures on
boxed plane partitions. Elementary Markov steps change the size of the
box from
(a x b x c) to ((a-1) x (b+1) x c) or ((a+1) x (b-1) x c). Algorithmic
realization of each step involves O((a+b)c) operations. One
application is an
efficient perfect random sampling algorithm for uniformly distributed
boxed
plane partitions.
Trajectories of our Markov chains can be viewed as random point
configurations in the three-dimensional lattice. We compute the bulk
limits of
the correlation functions of the resulting random point process on
suitable
two-dimensional sections. The limiting correlation functions define a
two-dimensional determinantal point processes with certain Gibbs
properties.
http://arxiv.org/abs/0804.3071
---------------------------------------------------------------
6960. INTERLACED PROCESSES ON THE CIRCLE
Anthony P. Metcalfe and Neil O'Connell and Jon Warren
When two Markov operators commute, it suggests that we can couple two
copies
of one of the corresponding processes. We explicitly construct a
number of
couplings of this type for a commuting family of Markov processes on
the set of
conjugacy classes of the unitary group, using a dynamical rule
inspired by the
RSK algorithm. Our motivation for doing this is to develop a parallel
programme, on the circle, to some recently discovered connections in
random
matrix theory between reflected and conditioned systems of particles
on the
line. One of the Markov chains we consider gives rise to a family of
Gibbs
measures on `bead configurations' on the infinite cylinder. We show
that these
measures have determinantal structure and compute the corresponding
space-time
correlation kernel.
http://arxiv.org/abs/0804.3142
---------------------------------------------------------------
6961. CRAM\'{E}R ASYMPTOTICS FOR FINITE TIME FIRST PASSAGE
PROBABILITIES OF GENERAL L\'{E}VY PROCESSES
Zbigniew Palmowski and Martijn Pistorius
We derive the exact asymptotics of $P(\sup_{u\leq t}X(u) > x)$ if $x$
and $t$
tend to infinity with $x/t$ constant, for a L\'{e}vy process $X$ that
admits
exponential moments. The proof is based on a renewal argument and a
two-dimensional renewal theorem of H\"{o}glund (1990).
http://arxiv.org/abs/0804.3169
---------------------------------------------------------------
6962. CONVEX RISK MEASURES: LEBESGUE PROPERTY ON ONE PERIOD AND MULTI
PERIOD RISK MEASURES AND APPLICATION IN CAPITAL ALLOCATION PROBLEM
Hirbod Assa
In this work we study the Lebesgue property for convex risk measures
on the
space of bounded c\`adl\`ag random processes ($\mathcal{R}^\infty$).
Lebesgue
property has been defined for one period convex risk measures in
\cite{Jo} and
earlier had been studied in \cite{De} for coherent risk measures. We
introduce
and study the Lebesgue property for convex risk measures in the multi
period
framework. We give presentation of all convex risk measures with
Lebesgue
property on bounded c\`adl\`ag processes. To do that we need to have a
complete
description of compact sets of $\mathcal{A}^1$. The main mathematical
contribution of this paper is the characterization of the compact sets
of
$\mathcal{A}^p$ (including $\mathcal{A}^1$). At the final part of this
paper,
we will solve the Capital Allocation Problem when we work with
coherent risk
measures.
http://arxiv.org/abs/0804.3209
---------------------------------------------------------------
6963. OPTIMAL STOPPING FOR L\'EVY PROCESSES AND AFFINE FUNCTIONS
Diana Dorobantu (LSProba)
This paper studies an optimal stopping problem for L\'evy processes.
We give
a justification of the form of the Snell envelope using standard
results of
optimal stopping. We also justify the convexity of the value function,
and
without a priori restriction to a particular class of stopping times,
we deduce
that the smallest optimal stopping time is necessarily a hitting time.
We
propose a method which allows to obtain the optimal threshold.
Moreover this
method allows to avoid long calculations of the integro-differential
operatorused in the usual proofs.
http://arxiv.org/abs/0804.3277
---------------------------------------------------------------
6964. THE ANTI-SYMMETRIC GUE MINOR PROCESS
Peter J. Forrester and Eric Nordenstam
Our study is initiated by a multi-component particle system underlying
the
tiling of a half hexagon by three species of rhombi. In this particle
system
species $j$ consists of $\lfloor j/2 \rfloor$ particles which are
interlaced
with neigbouring species. The joint probability density function (PDF)
for this
particle system is obtained, and is shown in a suitable scaling limit to
coincide with the joint eigenvalue PDF for the process formed by the
successive
minors of anti-symmetric GUE matrices, which in turn we compute from
first
principles. The correlations for this process are determinantal and we
give an
explicit formula for the corresponding correlation kernel in terms of
Hermite
polynomials. Scaling limits of the latter are computed, giving rise to
the Airy
kernel, extended Airy kernel and bead kernel at the soft edge and in
the bulk,
as well as a new kernel at the hard edge.
http://arxiv.org/abs/0804.3293
---------------------------------------------------------------
6965. A PROOF OF THE DALANG-MORTON-WILLINGER THEOREM
Dmitry B. Rokhlin
We give a new proof of the Dalang-Morton-Willinger theorem, relating the
no-arbitrage condition in stochastic securities market models to the
existence
of an equivalent martingale measure with bounded density for a $d$-
dimensional
stochastic sequence $(S_n)_{n=0}^N$ of stock prices. Roughly speaking,
the
proof is reduced to the assertion that under the no-arbitrage
condition for N=1
and $S\in L^1$ there exists a strictly positive linear fucntional on
$L^1$,
which is bounded from above on a special subset of the subspace $K
\subset L^1$
of investor's gains.
http://arxiv.org/abs/0804.3308
---------------------------------------------------------------
6966. SCALING LIMITS OF A TAGGED PARTICLE IN THE EXCLUSION PROCESS
WITH VARIABLE DIFFUSION COEFFICIENT
Milton Jara and Patricia Goncalves
We prove a law of large numbers and a central limit theorem for a tagged
particle in a symmetric simple exclusion process in the one-
dimensional lattice
with variable diffusion coefficient. The scaling limits are obtained
from a
similar result for the current through -1/2 for a zero-range process
with bond
disorder. For the CLT, we prove convergence to a fractional Brownian
motion of
Hurst exponent 1/4.
http://arxiv.org/abs/0804.3018
---------------------------------------------------------------
6967. ANISOTROPIC GROWTH OF RANDOM SURFACES IN 2+1 DIMENSIONS
Patrik L. Ferrari (1) and Alexei Borodin (2) ((1) WIAS-Berlin and
(2) Caltech)
We construct a family of stochastic growth models in 2+1 dimensions,
that
belong to the anisotropic KPZ class. Appropriate projections of these
models
yield 1+1 dimensional growth models in the KPZ class and random tiling
models.
We show that correlation functions associated to our models have
determinantal
structure, and we study large time asymptotics for one of the models.
The main asymptotic results are: (1) The growing surface has a
limit shape
that consists of facets interpolated by a curved piece. (2) The one-
point
fluctuations of the height function in the curved part are
asymptotically
normal with variance of order ln(t) for time t>>1. (3) There is a map
of the
(2+1)-dimensional space-time to the upper half-plane H such that on
space-like
submanifolds the multi-point fluctuations of the height function are
asymptotically equal to those of the pullback of the Gaussian free
(massless)
field on H.
http://arxiv.org/abs/0804.3035
---------------------------------------------------------------
6968. R-BOUNDEDNESS OF SMOOTH OPERATOR-VALUED FUNCTIONS
Mark Veraar and Tuomas Hytonen
In this paper we study $R$-boundedness of operator families
$\mathcal{T}\subset \calL(X,Y)$, where $X$ and $Y$ are Banach spaces.
Under
cotype and type assumptions on $X$ and $Y$ we give sufficient
conditions for
$R$-boundedness. In the first part we show that certain integral
operator are
$R$-bounded. This will be used to obtain $R$-boundedness in the case
that
$\mathcal{T}$ is the range of an operator-valued function $T:\R^d\to
\calL(X,Y)$ which is in a certain Besov space $B^{d/r}_{r,1}(\R^d;
\calL(X,Y))$.
The results will be applied to obtain $R$-boundedness of semigroups and
evolution families, and to obtain sufficient conditions for existence of
solutions for stochastic Cauchy problems.
http://arxiv.org/abs/0804.3313
---------------------------------------------------------------
6969. ARR\^ET OPTIMAL POUR LES PROCESSUS DE MARKOV FORTS ET LES
FONCTIONS AFFINES
Diana Dorobantu (LSProba)
In this Note we study optimal stopping problems for strong Markov
processes
and affine functions. We give a justification of the Snell envelope
form using
standard results of optimal stopping. We also justify the convexity of
the
value function, and without a priori restriction to a particular class
of
stopping times, we deduce that the smallest optimal stopping time is
necessarily a hitting time.
http://arxiv.org/abs/0804.3496
---------------------------------------------------------------
6970. RANDOM WALK IN DETERMINISTICALLY CHANGING ENVIRONMENT
Dmitry Dolgopyat and Carlangelo Liverani
We consider a random walk with transition probabilities weakly
dependent on
an environment with a deterministic, but strongly chaotic, evolution.
We prove
that for almost all initial conditions of the environment the walk
satisfies
the CLT.
http://arxiv.org/abs/0804.3497
---------------------------------------------------------------
6971. PHASE TRANSITION IN THE 1D RANDOM FIELD ISING MODEL WITH LONG
RANGE INTERACTION
Marzio Cassandro and Enza Orlandi and Pierre Picco (LATP)
We study the one dimensional Ising model with ferromagnetic, long range
interaction which decays as |i-j|^{-2+a}, 1/2< a<1, in the presence of
an
external random filed. we assume that the random field is given by a
collection
of independent identically distributed random variables, subgaussian
with mean
zero. We show that for temperature and strength of the randomness
(variance)
small enough with P=1 with respect to the distribution of the random
fields
there are at least two distinct extremal Gibbs measures.
http://arxiv.org/abs/0804.3672
---------------------------------------------------------------
6972. ON ESTIMATION OF FINITE POPULATION PROPORTION
Xinjia Chen
In this paper, we derive an explicit sample size formula for
estimating the
proportion of a finite population. The sample size obtained from the
formula
ensures a mixed criterion of absolute and relative errors.
http://arxiv.org/abs/0804.3779
---------------------------------------------------------------
6973. THE M/M/1 QUEUE IS BERNOULLI
Michael Keane and Neil O'Connell
The classical output theorem for the M/M/1 queue, due to Burke (1956),
states
that the departure process from a stationary M/M/1 queue, in
equilibrium, has
the same law as the arrivals process, that is, it is a Poisson
process. In this
paper we show that the associated measure-preserving transformation is
metrically isomorphic to a two-sided Bernoulli shift. We also discuss
some
extensions of Burke's theorem where it remains an open problem to
determine if,
or under what conditions, the analogue of this result holds.
http://arxiv.org/abs/0804.3935
---------------------------------------------------------------
6974. QUANTUM GROSS LAPLACIAN AND APPLICATIONS
Habib Ouerdiane and Samah Horrigue
In this paper, we introduce and study a noncommutative extension of
the Gross
Laplacian, called quantum Gross Laplacian. Then, applying the quantum
Gross
Laplacian to the particular case where the operator is the
multiplication
operator, we find a relation between classical and quantum Gross
Laplacian. As
application, we give explicit solution of linear quantum white noise
differential equation. In particular, we give a explicit solution of the
quantum Gross heat equation.
http://arxiv.org/abs/0804.3938
---------------------------------------------------------------
6975. THE NOISY VETO-VOTER MODEL: A RECURSIVE DISTRIBUTIONAL EQUATION
ON [0,1]
Saul Jacka and Marcus Sheehan
We study a particular example of a recursive distributional equation
(RDE) on
the unit interval. We identify all invariant distributions, the
corresponding
"basins of attraction" and address the issue of endogeny for the
associated
tree-indexed problem, making use of an extension of a recent result of
Warren.
http://arxiv.org/abs/0804.3943
---------------------------------------------------------------
6976. MAXIMUM PROBABILITY AND RELATIVE ENTROPY MAXIMIZATION. BAYESIAN
MAXIMUM PROBABILITY AND EMPIRICAL LIKELIHOOD
M. Grendar
Works, briefly surveyed here, are concerned with two basic methods:
Maximum
Probability and Bayesian Maximum Probability; as well as with their
asymptotic
instances: Relative Entropy Maximization and Maximum Non-parametric
Likelihood.
Parametric and empirical extensions of the latter methods - Empirical
Maximum
Maximum Entropy and Empirical Likelihood - are also mentioned. The
methods are
viewed as tools for solving certain ill-posed inverse problems, called
Pi-problem, Phi-problem, respectively. Within the two classes of
problems,
probabilistic justification and interpretation of the respective
methods are
discussed.
http://arxiv.org/abs/0804.3926
---------------------------------------------------------------
6977. APPROXIMATE CONTROLLABILITY FOR LINEAR STOCHASTIC DIFFERENTIAL
EQUATIONS IN INFINITE DIMENSIONS
D. Goreac
The objective of the paper is to investigate the approximate
controllability
property of a linear stochastic control system with values in a
separable real
Hilbert space. In a first step we prove the existence and uniqueness
for the
solution of the dual linear backward stochastic differential equation.
This
equation has the particularity that in addition to an unbounded
operator acting
on the Y-component of the solution there is still another one acting
on the
Z-component. With the help of this dual equation we then deduce the
duality
between approximate controllability and observability. Finally, under
the
assumption that the unbounded operator acting on the state process of
the
forward equation is an infinitesimal generator of an exponentially
stable
semigroup, we show that the generalized Hautus test provides a necessary
condition for the approximate controllability. The paper generalizes
former
results by Buckdahn, Quincampoix and Tessitore (2006) and Goreac
(2007) from
the finite dimensional to the infinite dimensional case.
http://arxiv.org/abs/0804.3893
---------------------------------------------------------------
6978. INSURANCE, REINSURANCE AND DIVIDEND PAYMENT
D. Goreac
The aim of this paper is to introduce an insurance model allowing
reinsurance
and dividend payment. Our model deals with several homogeneous
contracts and
takes into account the legislation regarding the provisions to be
justified by
the insurance companies. This translates into some restriction on the
(maximal)
number of contracts the company is allowed to cover. We deal with a
controlled
jump process in which one has free choice of retention level and
dividend
amount. The value function is given as the maximized expected discounted
dividends. We prove that this value function is a viscosity solution
of some
first-order Hamilton-Jacobi-Bellman variational inequality. Moreover, a
uniqueness result is provided.
http://arxiv.org/abs/0804.3900
---------------------------------------------------------------
6979. SUSPENSION FLOWS OVER VERSHIK'S AUTOMORPHISMS
Alexander I. Bufetov
A multiplicative asymptotics is obtained for the deviation of ergodic
averages for certain classes of suspension flows over Vershik's
automorphisms.
http://arxiv.org/abs/0804.3970
---------------------------------------------------------------
6980. A FAMILY OF SERIES REPRESENTATIONS OF THE MULTIPARAMETER
FRACTIONAL BROWNIAN MOTION
Anatoliy Malyarenko
We derive a family of series representations of the multiparameter
fractional
Brownian motion in the centred ball of radius $R$ in the $N$-
dimensional space
$\mathbb{R}^N$. Some known examples of series representations are
shown to be
the members of the family under consideration.
http://arxiv.org/abs/0804.4076
---------------------------------------------------------------
6981. LOGARITHMIC COMPONENTS OF THE VACANT SET FOR RANDOM WALK ON A
DISCRETE TORUS
David Windisch
This work continues the investigation, initiated in a recent work by
Benjamini and Sznitman, of percolative properties of the set of points
not
visited by a random walk on the discrete torus (Z/NZ)^d up to time
uN^d in high
dimension d. If u>0 is chosen sufficiently small it has been shown
that with
overwhelming probability this vacant set contains a unique giant
component
containing segments of length c_0 log N for some constant c_0 > 0, and
this
component occupies a non-degenerate fraction of the total volume as N
tends to
infinity. Within the same setup, we investigate here the complement of
the
giant component in the vacant set and show that some components
consist of
segments of logarithmic size. In particular, this shows that the
choice of a
sufficiently large constant c_0 > 0 is crucial in the definition of
the giant
component.
http://arxiv.org/abs/0804.4097
---------------------------------------------------------------
6982. GAUSSIAN LIMITS FOR GENERALIZED SPACINGS
Yu. Baryshnikov and Mathew D. Penrose and J. E. Yukich
Nearest neighbor cells in $R^d$ are used to define coefficients of
divergence
($\phi$-divergences) between continuous multivariate samples. For
large sample
sizes, such distances are shown to be asymptotically normal with a
variance
depending on the underlying point density. The finite-dimensional
distributions
of the point measures induced by the coefficients of divergence
converge to
those of a generalized Gaussian field with a covariance structure
determined by
the point densities. In $d = 1$, this extends classical central limit
theory
for sum functions of spacings. The general results yield central limit
theorems
for logarithmic $k$-spacings, information gain, log-likelihood ratios,
and the
number of pairs of sample points within a fixed distance of each other.
http://arxiv.org/abs/0804.4123
---------------------------------------------------------------
6983. CONTINUITY PROPERTIES AND INFINITE DIVISIBILITY OF STATIONARY
DISTRIBUTIONS OF SOME GENERALISED ORNSTEIN-UHLENBECK PROCESSES
Alexander Lindner and Ken-iti Sato
Properties of the law $\mu$ of the integral $\int_0^{\infty}c^{-
N_{t-}}dY_t$
are studied, where $c>1$ and $\{(N_t,Y_t), t\geq 0\}$ is a bivariate L
\'evy
process such that $\{N_t \}$ and $\{Y_t \}$ are Poisson processes with
parameters $a$ and $b$, respectively. This is the stationary
distribution of
some generalised Ornstein-Uhlenbeck process. The law $\mu$ is
parametrised by
$c$, $q$ and $r$, where $p=1-q-r$, $q$, and $r$ are the normalised L
\'evy
measure of $\{(N_t,Y_t)\}$ at the points $(1,0)$, $(0,1)$ and $(1,1)$,
respectively. It is shown that, under the condition that $p>0$ and
$q>0$,
$\mu_{c,q,r}$ is infinitely divisible if and only if $r\leq pq$. The
infinite
divisibility of the symmetrisation of $\mu$ is also characterised. The
law
$\mu$ is either continuous-singular or absolutely continuous, unless
$r=1$. It
is shown that if $c$ is in the set of Pisot-Vijayaraghavan numbers,
which
includes all integers bigger than 1, then $\mu$ is continuous-singular
under
the condition $q>0$. On the other hand, for Lebesgue almost every
$c>1$, there
are positive constants $C_1$ and $C_2$ such that $\mu$ is absolutely
continuous
whenever $q\geq C_1 p \geq C_2 r$. For any $c>1$, there is a positive
constant
$C_3$ such that $\mu$ is continuous-singular whenever $q>0$ and
$\max\{q,r\}\leq C_3 p$. Here, if $\{N_t \}$ and $\{Y_t \}$ are
independent,
then $r=0$ and $q=b/(a+b)$.
http://arxiv.org/abs/0804.4258
---------------------------------------------------------------
6984. LARGE DEVIATIONS FOR RANDOM SPECTRAL MEASURES AND SUM RULES
Fabrice Gamboa (IMT) and Alain Rouault (LMA-Versailles)
We prove a Large Deviation Principle for the random spec- tral measure
associated to the pair $(H_N; e)$ where $H_N$ is sampled in the GUE(N)
and e is
a fixed unit vector (and more generally in the $\beta$- extension of
this
model). The rate function consists of two parts. The contribution of the
absolutely continuous part of the measure is the reversed Kullback
information
with respect to the semicircle distribution and the contribution of the
singular part is connected to the rate function of the extreme
eigenvalue in
the GUE. This method is also applied to the Laguerre and Jacobi
ensembles, but
in thoses cases the expression of the rate function is not so explicit.
http://arxiv.org/abs/0804.4322
---------------------------------------------------------------
6985. ERGODIC OPTIMAL QUADRATIC CONTROL FOR AN AFFINE EQUATION WITH
STOCHASTIC AND STATIONARY COEFFICIENTS
Giuseppina Guatteri and Federica Masiero
We study ergodic quadratic optimal stochastic control problems for an
affine
state equation with state and control dependent noise and with
stochastic
coefficients. We assume stationarity of the coefficients and a finite
cost
condition. We first treat the stationary case and we show that the
optimal cost
corresponding to this ergodic control problem coincides with the one
corresponding to a suitable stationary control problem and we provide
a full
characterization of the ergodic optimal cost and control.
http://arxiv.org/abs/0804.4362
---------------------------------------------------------------
6986. FRACTIONAL BROWNIAN FLOWS
Sreekar Vadlamani
We consider stochastic flow on n-dimensional Euclidean space driven by
fractional Brownian motion with Hurst parameter H greater than half,
and study
tangent flow and the growth of the Hausdorff measure of sub-manifolds
of the
ambient n-dimensional Euclidean space, as they evolve under the flow.
The main
result is a bound on the rate of (global) growth in terms of the
(local) Holder
norm of the flow.
http://arxiv.org/abs/0804.4376
---------------------------------------------------------------
6987. EXISTENCE AND REGULARITY OF A NONHOMOGENEOUS TRANSITION MATRIX
UNDER MEASURABILITY CONDITIONS
Liuer Ye (The School of Mathematics and Computational Science) and
Xianping Guo (The School of Mathematics and Computational Science), On
\'esimo
Hern\'andez-Lerma (Departamento de Matem\'aticas, CINVESTAV-IPN)
This paper is about the existence and regularity of the transition
probability matrix of a nonhomogeneous continuous-time Markov process
with a
countable state space. A standard approach to prove the existence of
such a
transition matrix is to begin with a continuous (in t) and
conservative matrix
Q(t)=[q_{ij}(t)] of nonhomogeneous transition rates q_{ij}(t), and use
it to
construct the transition probability matrix. Here we obtain the same
result
except that the q_{ij}(t) are only required to satisfy a mild
measurability
condition, and Q(t) may not be conservative. Moreover, the resulting
transition
matrix is shown to be the minimum transition matrix and, in addition, a
necessary and sufficient condition for it to be regular is obtained.
These
results are crucial in some applications of nonhomogeneous continuous-
time
Markov processes, such as stochastic optimal control problems and
stochastic
games, which motivated this work in the first place.
http://arxiv.org/abs/0804.4441
---------------------------------------------------------------
6988. CIRCULAR JACOBI ENSEMBLES AND DEFORMED VERBLUNSKY COEFFICIENTS
Paul Bourgade and Ashkan Nikeghbali and Alain Rouault
Using spectral theory of unitary operators and the theory of orthogonal
polynomials on the unit circle, we propose a simple matrix model for the
following circular analogue of the Jacobi ensemble:
$$c_{\delta,\beta}^{(n)} \prod_{1\leq k<l\leq n}
|e^{\ii\theta_k}-e^{\ii\theta_l}|^\beta
\prod_{j=1}^{n}(1-e^{-\ii\theta_j})^{\delta}
(1-e^{\ii\theta_j})^{\bar{\delta}}$$
with $\Re \delta > -1/2$. If $e$ is a cyclic vector for a unitary
$n\times n$ matrix $U$, the spectral measure of the pair $(U,e)$ is well
parameterized by its Verblunsky coefficients $
(\alpha_0,...,\alpha_{n-1})$.
We introduce here a deformation $(\gamma_0,...,\gamma_{n-1})$ of these
coefficients so that the associated Hessenberg matrix (called GGT) can
be
decomposed into a product $r(\gamma_0)... r(\gamma_{n-1})$ of elementary
reflections parameterized by these coefficients. If $\gamma_0,...,
\gamma_{n-1}$ are independent random variables with some remarkable
distributions, then the eigenvalues of the GGT matrix follow the
circular
Jacobi distribution above.
These deformed Verblunsky also allow to prove that, in the regime $
\delta =
\delta(n)$ with $\delta(n)/n \to \dd$, the spectral measure and the
empirical
spectral distribution weakly converge to an explicit nontrivial
probability
measure supported by an arc of the unit circle.
http://arxiv.org/abs/0804.4512
---------------------------------------------------------------
6989. CORDES CONDITIONS AND SOME ALTERNATIVES FOR PARABOLIC EQUATIONS
AND DISCONTINUOUS DIFFUSION
Nikolai Dokuchaev
The paper considers parabolic equations in non-divergent form with
discontinuous coefficients at higher derivatives. Their investigation
is most
complicated because, in general, in the case of discontinuous
coefficients, the
uniqueness of a solution for nonlinear parabolic or elliptic equations
can
fail, and there is no a priory estimate for partial derivatives of a
solution.
In this paper, existence and regularity results are obtained under
some Cordes
type restrictions on the coefficients. The results are applied to
diffusion
processes.
http://arxiv.org/abs/0804.4519
---------------------------------------------------------------
6990. OPTIMAL SOLUTION OF INVESTMENT PROBLEMS VIA LINEAR PARABOLIC
EQUATIONS GENERATED BY KALMAN FILTER
Nikolai Dokuchaev
We consider optimal investment problems for a diffusion market model
with
non-observable random drifts that evolve as an Ito's process. Admissible
strategies do not use direct observations of the market parameters,
but rather
use historical stock prices. For a non-linear problem with a general
performance criterion, the optimal portfolio strategy is expressed via
the
solution of a scalar minimization problem and a linear parabolic
equation with
coefficients generated by the Kalman filter.
http://arxiv.org/abs/0804.4522
---------------------------------------------------------------
6991. ISING MODELS ON LOCALLY TREE-LIKE GRAPHS
Amir Dembo and Andrea Montanari
We consider Ising models on graphs that converge locally to trees.
Examples
include random regular graphs with bounded degree and uniformly random
graphs
with bounded average degree. We prove that the `cavity' prediction for
the
limiting free energy per spin is correct for any temperature and
external
field. Further, local marginals can be approximated by iterating a set
of mean
field (cavity) equations. Both results are achieved by proving the local
convergence of the Boltzmann distribution on the original graph to the
Boltzmann distribution on the appropriate infinite random tree.
http://arxiv.org/abs/0804.4726
---------------------------------------------------------------
6992. THE SPECTRUM OF THE RANDOM ENVIRONMENT AND LOCALIZATION OF NOISE
Dimitrios Cheliotis and Balint Virag
We consider random walk on a mildly random environment on finite
transitive
d- regular graphs of increasing girth. After scaling and centering, the
analytic spectrum of the transition matrix converges in distribution
to a
Gaussian noise. An interesting phenomenon occurs at d = 2: as the
limit graph
changes from a regular tree to the integers, the noise becomes
localized.
http://arxiv.org/abs/0804.4814
---------------------------------------------------------------
6993. HAMILTONICITY THRESHOLDS IN ACHLIOPTAS PROCESSES
Michael Krivelevich and Eyal Lubetzky and Benny Sudakov
In this paper we analyze the appearance of a Hamilton cycle in the
following
random process. The process starts with an empty graph on n labeled
vertices.
At each round we are presented with K=K(n) edges, chosen uniformly at
random
from the missing ones, and are asked to add one of them to the current
graph.
The goal is to create a Hamilton cycle as soon as possible.
We show that this problem has three regimes, depending on the value
of K. For
K=o(\log n), the threshold for Hamiltonicity is (1+o(1))n\log n /(2K),
i.e.,
typically we can construct a Hamilton cycle K times faster that in the
usual
random graph process. When K=\omega(\log n) we can essentially waste
almost no
edges, and create a Hamilton cycle in n+o(n) rounds with high
probability.
Finally, in the intermediate regime where K=\Theta(\log n), the
threshold has
order n and we obtain upper and lower bounds that differ by a
multiplicative
factor of 3.
http://arxiv.org/abs/0804.4707
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