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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
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