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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.
Author(s): Tanja Gernhard
Abstract: 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
Author(s): Jakub Onufry Wojtaszczyk
Abstract: 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
Author(s): Marcin Pilipczuk and Jakub Onufry Wojtaszczyk
Abstract: 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
Author(s): Yeneng Sun and Yongchao Zhang
Abstract: 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
Author(s): Ivan Nourdin (PMA) and Giovanni Peccati (LSTA)
Abstract: 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
Author(s): Massimiliano Gubinelli and Samy Tindel
Abstract: 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
Author(s): Joan-Andreu L\'azaro-Cam\'i and Juan-Pablo Ortega
Abstract: 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
Author(s): Iain MacPhee and Mikhail Menshikov and Dimitri Petritis and Serguei Popov
Abstract: 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
Author(s): Amos Nathan
Abstract: 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
Author(s): Mark Veraar
Abstract: 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
Author(s): U. Haboeck
Abstract: 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
Author(s): Josep Llu\'is Sol\'e and Frederic Utzet
Abstract: 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
Author(s): Min Song
Abstract: 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
Author(s): F. G\"otze and H. K\"osters
Abstract: 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
Author(s): H. K\"osters
Abstract: 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
Author(s): Marc Hoffmann (LAMA) and Nathalie Krell (PMA)
Abstract: 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
Author(s): Florian Sebert and Leslie Ying and and Yi Ming Zou
Abstract: 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
Author(s): Antonio Di Crescenzo and Barbara Martinucci
Abstract: 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
Author(s): Svante Janson
Abstract: 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
Author(s): Gautam Iyer and Jonathan Mattingly
Abstract: 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
Author(s): R. Spjut
Abstract: 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
Author(s): Antonio Di Crescenzo
Abstract: 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
Author(s): Peter Major
Abstract: 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
Author(s): Alexander Drewitz
Abstract: 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
Author(s): Michael Baake (Bielefeld) and Matthias Birkner (Berlin) and Robert V. Moody (Victoria)
Abstract: 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
Author(s): Shui Feng and Byron Schmuland and Jean Vaillancourt and and Xiaowen Zhou
Abstract: 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
Author(s): Olivier Raimond (LM-Orsay) and Bruno Schapira (LM-Orsay)
Abstract: 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
Author(s): Katrin Gelfert
Abstract: 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
Author(s): Bruno Schapira (LM-Orsay)
Abstract: 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
Author(s): T. Bodineau and G. Giacomin and H. Lacoin and F. Toninelli
Abstract: 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
Author(s): S.Hamad\'ene and H.Wang
Abstract: 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
Author(s): Constantinos Kardaras
Abstract: 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
Author(s): Ioannis Karatzas and Constantinos Kardaras
Abstract: 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
Author(s): Remco van der Hofstad and Mark Holmes
Abstract: 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
Author(s): Constantinos Kardaras and Eckhard Platen
Abstract: 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
Author(s): Pierre Andreoletti (MAPMO)
Abstract: 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
Author(s): Anders Karlsson and Nicolas Monod
Abstract: 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
Author(s): Jose A. Ramirez and Brian Rider
Abstract: 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
Author(s): Iosif Pinelis
Abstract: 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
Author(s): Constantinos Kardaras
Abstract: 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
Author(s): Michail Anthropelos and Gordan Zitkovic
Abstract: 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
Author(s): Dapeng Zhan
Abstract: 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
Author(s): Pedro Lei and David Nualart
Abstract: 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
Author(s): Dimitris Achlioptas and Amin Coja-Oghlan
Abstract: 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
Author(s): Michael Caruana and Peter Friz
Abstract: 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
Author(s): Z. Jiang and M.R. Pistorius
Abstract: 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
Author(s): Alexey M.Kulik
Abstract: 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
Author(s): Florin Avram (LMA-PAU) and Nikolai Leonenko and Ludmila Sakhno
Abstract: 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
Author(s): Pascal Moyal
Abstract: 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
Author(s): S\'andor Baran and Gyula Pap
Abstract: 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
Author(s): Alexandre Baraviera and Ruy Exel and Artur O. Lopes
Abstract: 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
Author(s): E.A. Pechersky and N.D. Vvedenskaya
Abstract: 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
Author(s): Clement Pellegrini (ICJ)
Abstract: "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
Author(s): Clement Pellegrini (ICJ)
Abstract: "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
Author(s): Shige Peng
Abstract: 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
Author(s): Andrei A. Fedorenko and Pierre Le Doussal and Kay Joerg Wiese
Abstract: 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
Author(s): Jens Christian Claussen
Abstract: 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
Author(s): Steven Finch and Zai-Qiao Bai and Pascal Sebah
Abstract: 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
Author(s): A. De Masi and I. Merola and E. Presutti and Y. Vignaud
Abstract: 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
Author(s): Yinfang Shen and Weian Zheng
Abstract: 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
Author(s): Balint Toth and Balint Veto
Abstract: 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
Author(s): David F. Anderson and Gheorghe Craciun and Thomas G. Kurtz
Abstract: 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
Author(s): Ameur Dhahri (CEREMADE)
Abstract: 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
Author(s): Ameur Dhahri (ICJ and Ceremade)
Abstract: 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
Author(s): Joachim Krug
Abstract: 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
Author(s): Laszlo Erdos and Benjamin Schlein and Horng-Tzer Yau
Abstract: 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
Author(s): Thomas Kriecherbauer and Joachim Krug
Abstract: 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
Author(s): N. Fountoulakis and D. K\"uhn and D. Osthus
Abstract: 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
Author(s): Robert Fernholz and Ioannis Karatzas and Constantinos Kardaras
Abstract: 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
Author(s): Jim Pitman and Matthias Winkel
Abstract: 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
Author(s): Rados{\l}aw Adamczak
Abstract: 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
Author(s): Richard F. Bass and Moritz Kassmann and and Takashi Kumagai
Abstract: 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
Author(s): C. Berg and C. Vignat
Abstract: 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
Author(s): Anne-Laure Basdevant and Arvind Singh
Abstract: 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
Author(s): Elchanan Mossel (UC Berkeley) and Arnab Sen (UC Berkeley)
Abstract: 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
Author(s): George Lowther
Abstract: 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
Author(s): Guangyue Han and Brian Marcus
Abstract: 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
Author(s): Iain M. Johnstone
Abstract: 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
Author(s): K. Bahlali (IMATH) and Abouo Elouaflin (UFR-MI) and E. Pardoux (CMI)
Abstract: 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
Author(s): Arup Bose and Amites Dasgupta and Krishanu Maulik
Abstract: 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
Author(s): Xavier Bardina and Carles Rovira
Abstract: 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
Author(s): Alexander Weiss
Abstract: 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
Author(s): Dmitri Finkelshtein and Yuri Kondratiev and Eugene Lytvynov
Abstract: 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
Author(s): Guangyan Jia and Zhiyong Yu
Abstract: 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
Author(s): Litan Yan and Xiangfeng Yang
Abstract: In this paper, we study the stochastic integration with respect to local
times of fractional Brownian motion $B^H$ with Hurst index $1/2
http://arxiv.org/abs/0803.3665
Author(s): Akimichi Takemura and Vladimir Vovk and and Glenn Shafer
Abstract: 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
Author(s): Mingyu Xu
Abstract: 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
Author(s): Gerold Alsmeyer and Alex Iksanov and Uwe Roesler
Abstract: 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
Author(s): Ross Pinsky
Abstract: 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|
http://arxiv.org/abs/0803.3748
Author(s): Christophe Garban and Gabor Pete and Oded Schramm
Abstract: 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
Author(s): Paul Bourgade
Abstract: 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
Author(s): Federico Camia and Matthijs Joosten and Ronald Meester
Abstract: 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
Author(s): Kieran Kelly and Przemyslaw Repetowicz and Seosamh macReamoinn
Abstract: 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
Author(s): Alexander Barvinok and Zur Luria and Alex Samorodnitsky and and Alexander Yong
Abstract: 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
Author(s): Alexei Onatski
Abstract: 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
http://arxiv.org/abs/0803.4155
Author(s): F.Aurzada and I.A.Ibragimov and M.A.Lifshits and J.H. van Zanten
Abstract: 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
Author(s): G. Morvai and B. Weiss
Abstract: 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
Author(s): Paolo Guasoni and Mikl\'os R\'asonyi and Walter Schachermayer
Abstract: 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
Author(s): F. Cipriano and H. Ouerdiane and R. Vilela Mendes
Abstract: 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
Author(s): Steve Zelditch
Abstract: 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
Author(s): Leonid Bogachev and Alexei Daletskii
Abstract: 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
Author(s): Erlin Yao
Abstract: 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
Author(s): Pawe{\l} Sztonyk
Abstract: 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
Author(s): Gerald Trutnau
Abstract: 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
Author(s): Carlos M. Mora and Rolando Rebolledo
Abstract: 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
Author(s): Gerald Trutnau
Abstract: 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
Author(s): Johannes Leitner
Abstract: 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
Author(s): Oskar Sandberg
Abstract: 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
Author(s): J. Theodore Cox and Edwin A. Perkins
Abstract: 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
Author(s): L. C. G. Rogers and Fanyin Zhou
Abstract: 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
Author(s): Daniela Bertacchi and Fabio Zucca
Abstract: 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
Author(s): Gregory Schehr and Satya N. Majumdar
Abstract: 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
Author(s): Atilla Yilmaz
Abstract: 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
Author(s): Z. Brzezniak and J. M. A. M. van Neerven and M. C. Veraar and L. Weis
Abstract: 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
Author(s): Kumiko Hattori and Tetsuya Hattori
Abstract: 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
Author(s): Igor Cialenco and Sergey Lototsky and Jan Pospisil
Abstract: 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
Author(s): Rainer Buckdahn and Juan Li
Abstract: 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
Author(s): Steven N. Evans and Alex Gottlieb
Abstract: 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
Author(s): Antonis Papapantoleon
Abstract: 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
Author(s): Stefan Gerhold and Martin Zeiner
Abstract: 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
Author(s): Oskar Sandberg
Abstract: 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
Author(s): Emanuel Milman
Abstract: 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
Author(s): Guillaume Chapuy
Abstract: 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
Author(s): Jean Bertoin (DMA and PMA) and Vladas Sidoravicius (UCI and CWI) and Maria Eulalia Vares (CBPF)
Abstract: 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
Author(s): Arkadius Kalka and Mina Teicher and and Boaz Tsaban
Abstract: 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
Author(s): Clemens Heuberger and Helmut Prodinger
Abstract: 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
Author(s): Nedzad Limi\'c
Abstract: 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
Author(s): Vincent Bansaye (PMA)
Abstract: 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
Author(s): Bela Bollobas and Oliver Riordan
Abstract: 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
Author(s): Franck Barthe (IMT) and Nolwen Huet (IMT)
Abstract: 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
Author(s): Dong Wang
Abstract: 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
Author(s): Gerold Alsmeyer and Alexander Iksanov
Abstract: 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
Author(s): Mauro Mariani
Abstract: 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
Author(s): Romain Abraham (MAPMO) and Jean-Francois Delmas (CERMICS) and Guillaume Voisin (MAPMO)
Abstract: 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
Author(s): Peter Spreij and Enno Veerman and Peter Vlaar
Abstract: 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
Author(s): Marta Sanz-Sol\'e and Iv\'an Torrecilla-Tarantino
Abstract: 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
Author(s): J. M. A. M. van Neerven and M. C. Veraar and L. Weis
Abstract: 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
Author(s): Quang-Cuong Pham
Abstract: 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
Author(s): L. Cecconi and A. Gandolfi and C. C. A. Sastri
Abstract: 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
Author(s): N. Haydn and S. Vaienti
Abstract: 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
Author(s): V.Vatutin and A.E.Kyprianou
Abstract: 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
Author(s): Michael Scheutzow
Abstract: 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
Author(s): Sergio De Carvalho Bezerra
Abstract: 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
Author(s): Arnaud Debussche (IRMAR)
Abstract: 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
Author(s): Craig A. Tracy and Harold Widom
Abstract: 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
Author(s): Marton Balazs and Julia Komjathy
Abstract: 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
Author(s): V. V. Yurinsky
Abstract: 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
Author(s): Thomas Mountford and Pierre Tarr\`es
Abstract: 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
Author(s): Jeffrey M. Rosenbluth
Abstract: 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
Author(s): Driss Gretete
Abstract: 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
Author(s): Xinjia Chen
Abstract: 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
Author(s): Jan Maas and Jan van Neerven
Abstract: 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
http://arxiv.org/abs/0804.1432
Author(s): Michel Weber
Abstract: We give several applications of an identity for sums of weakly stationary
sequences due to Ky Fan.
http://arxiv.org/abs/0804.1508
Author(s): Lincoln Chayes and Nicholas Crawford and Dmitry Ioffe and and Anna Levit
Abstract: 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
Author(s): Damien Bankovsky and Allan Sly
Abstract: 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
Author(s): Jir\^o Akahori and Yuuki Kanashi and Yuichi Morimura
Abstract: 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
Author(s): Svante Janson
Abstract: 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
Author(s): Dmitry B. Rokhlin
Abstract: 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
Author(s): Pablo A. Ferrari and Patricia Goncalves and James B. Martin
Abstract: 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
Author(s): Sergio Albeverio and Olga Rozanova
Abstract: 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
Author(s): Leonardo T. Rolla and Augusto Q. Teixeira
Abstract: 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
Author(s): Olivier Durieu
Abstract: 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
Author(s): Frank Aurzada and Mikhail Lifshits and Werner Linde
Abstract: 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
Author(s): Gerold Alsmeyer and Uwe R\"osler
Abstract: 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
Author(s): Ivan Nourdin (PMA) and Giovanni Peccati (LSTA) and Anthony R\'eveillac (LMA-Rochelle)
Abstract: 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
Author(s): Tomaz Podobnik and Tomi Zivko
Abstract: 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
Author(s): Dieter Denneberg and Michel Grabisch (LIP6)
Abstract: 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
Author(s): Michel Grabisch (LIP6)
Abstract: 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
Author(s): Richard F. Bass and Krzysztof Burdzy and Zhen-Qing Chen and Martin Hairer
Abstract: 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
Author(s): Lung-Chi Chen and Akira Sakai
Abstract: 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
Author(s): Antonio Galves and Eva L\"ocherbach
Abstract: 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
Author(s): Marc Wouts (MODAL'x)
Abstract: 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
Author(s): Henri Comman
Abstract: 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
Author(s): Thierry L\'evy (DMA)
Abstract: 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
Author(s): J. Lember and A. Koloydenko
Abstract: 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
Author(s): Henri Comman
Abstract: 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
Author(s): Martin Haenggi
Abstract: 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
Author(s): Vyacheslav M. Abramov
Abstract: 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
Author(s): S. R. S. Varadhan
Abstract: 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
Author(s): 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)
Abstract: 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
Author(s): Elise Janvresse (LMRS) and Beno\^it Rittaud (IG and LMPT) and Thierry De La Rue (LMRS)
Abstract: 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
http://arxiv.org/abs/0804.2378
Author(s): Elise Janvresse (LMRS) and Beno\^it Rittaud (IG and LMPT) and Thierry De La Rue (LMRS)
Abstract: 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
Author(s): Lorenzo Bertini and Claudio Landim and Mustapha Mourragui
Abstract: 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
Author(s): T. Tao and V. Vu
Abstract: 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
Author(s): Jean-Christophe Breton (LMA-Rochelle) and Ivan Nourdin (PMA)
Abstract: 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
Author(s): Nicole El Karoui and Asma Meziou
Abstract: 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
Author(s): Amaury Lambert (CMAP and Fese)
Abstract: 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
Author(s): Josep Llu\'is Sol\'e and Frederic Utzet
Abstract: 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
Author(s): Allan Sly and Chris Heyde
Abstract: 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
Author(s): Amine Asselah
Abstract: 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
Author(s): Elena R. Loubenets
Abstract: 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
Author(s): Alexander Schoenhuth
Abstract: 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
Author(s): J.-R. Chazottes and Z. Coelho and P. Collet
Abstract: 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
Author(s): J.-R. Chazottes and Z. Coelho and P. Collet
Abstract: 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
Author(s): Hirbod Assa
Abstract: 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
Author(s): Ramon van Handel
Abstract: 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
Author(s): Constantinos Kardaras
Abstract: 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
Author(s): Jean-Francois Le Gall
Abstract: 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
Author(s): Nicolas Fournier and Jacques Printems
Abstract: 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
Author(s): Alexander Gnedin and Alex Iksanov and Uwe Roesler
Abstract: 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
Author(s): Pierre Calka (MAP5) and Julien Michel (UMPA-ENSL) and Katy Paroux (LM-Besan\c{c}on, IRISA)
Abstract: 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
Author(s): Nathael Gozlan (LAMA)
Abstract: 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
Author(s): Xinjia Chen
Abstract: 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
Author(s): Shuhei Mano
Abstract: 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
Author(s): Michel Grabisch (CES) and Christophe Labreuche (TRT)
Abstract: 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
Author(s): Ana Cristina Moreira Freitas and Jorge Milhazes Freitas and Mike Todd
Abstract: 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
Author(s): Alexei Borodin and Vadim Gorin
Abstract: 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
Author(s): Anthony P. Metcalfe and Neil O'Connell and Jon Warren
Abstract: 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
Author(s): Zbigniew Palmowski and Martijn Pistorius
Abstract: 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
Author(s): Hirbod Assa
Abstract: 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
Author(s): Diana Dorobantu (LSProba)
Abstract: 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
Author(s): Peter J. Forrester and Eric Nordenstam
Abstract: 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
Author(s): Dmitry B. Rokhlin
Abstract: 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
Author(s): Milton Jara and Patricia Goncalves
Abstract: 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
Author(s): Patrik L. Ferrari (1) and Alexei Borodin (2) ((1) WIAS-Berlin and (2) Caltech)
Abstract: 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
Author(s): Mark Veraar and Tuomas Hytonen
Abstract: 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
Author(s): Diana Dorobantu (LSProba)
Abstract: 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
Author(s): Dmitry Dolgopyat and Carlangelo Liverani
Abstract: 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
Author(s): Marzio Cassandro and Enza Orlandi and Pierre Picco (LATP)
Abstract: 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
Author(s): Xinjia Chen
Abstract: 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
Author(s): Michael Keane and Neil O'Connell
Abstract: 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
Author(s): Habib Ouerdiane and Samah Horrigue
Abstract: 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
Author(s): Saul Jacka and Marcus Sheehan
Abstract: 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
Author(s): M. Grendar
Abstract: 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
Author(s): D. Goreac
Abstract: 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
Author(s): D. Goreac
Abstract: 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
Author(s): Alexander I. Bufetov
Abstract: 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
Author(s): Anatoliy Malyarenko
Abstract: 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
Author(s): David Windisch
Abstract: 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
Author(s): Yu. Baryshnikov and Mathew D. Penrose and J. E. Yukich
Abstract: 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
Author(s): Alexander Lindner and Ken-iti Sato
Abstract: 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
Author(s): Fabrice Gamboa (IMT) and Alain Rouault (LMA-Versailles)
Abstract: 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
Author(s): Giuseppina Guatteri and Federica Masiero
Abstract: 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
Author(s): Sreekar Vadlamani
Abstract: 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
Author(s): 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)
Abstract: 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
Author(s): Paul Bourgade and Ashkan Nikeghbali and Alain Rouault
Abstract: 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 -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
Author(s): Nikolai Dokuchaev
Abstract: 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
Author(s): Nikolai Dokuchaev
Abstract: 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
Author(s): Amir Dembo and Andrea Montanari
Abstract: 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
Author(s): Dimitrios Cheliotis and Balint Virag
Abstract: 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
Author(s): Michael Krivelevich and Eyal Lubetzky and Benny Sudakov
Abstract: 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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