Probability Abstracts 83
This document contains abstracts 2825-2956.
They have been mailed on October 30, 2004.
2825. A LOWER BOUND FOR THE CHEMICAL DISTANCE IN SPARSE LONG-RANGE PERCOLATION
MODELS
Noam Berger
We consider long-range percolation in dimension $d\geq 1$, where distinct
sites $x$ and $y$ are connected with probability $p_{x,y}\in[0,1]$. Assuming
that $p_{x,y}$ is translation invariant and that $p_{x,y}=\|x-y\|^{-s+o(1)}$
with $s>2d$, we show that the graph distance is at least linear with the
Euclidean distance.
2826.
ON NUMERICAL SOLUTIONS TO STOCHASTIC VOLTERRA EQUATIONS
Anna Karczewska and Piotr Rozmej
The aim of the paper is to demonstrate the use of the Galerkin method for
some kind of Volterra equations, determininistic and stochastic as well.
The paper consists of two parts: the theoretical and numerical one.
In the first part we recall some apparently well-known results concerning the
Volterra equations under consideration. In the second one we describe a
numerical algorithm used and next present some examples of numerical solutions
in order to illustrate the pertinent features of the technique used in the
paper.
2827.
WHY THERE ARE NO GAPS IN THE SUPPORT OF NON-NEGATIVE INTEGER-VALUED
INFINITELY DIVISIBLE LAWS?
S. Satheesh
Remark.9 in Bose-Dasgupta-Rubin (2002) review states that when a non-negative
integer-valued infinitely divisible law has an atom at unity then its support
cannot have any gaps. Here one has two questions. (i) Why there are no gaps and
(ii) Can there be gaps if the condition is not satisfied. Our investigation
with these questions in mind centers on the implications of having and not
having atoms at zero and unity. We give two examples/ constructions, which show
that the remark needs modification and we modify it.
2828.
AN ANALYSIS OF ISING TYPE MODELS ON CAYLEY TREE BY A CONTOUR ARGUMENT
U.A.Rozikov
In the paper the Ising model with competing $J_1$ and $J_2$ interactions with
spin values $\pm 1$, on a Cayley tree of order 2 (with 3 neighbors) is
considered . We study the structure of the ground states and verify the Peierls
condition for the model. Our second result gives description of Gibbs measures
for ferromagnetic Ising model with $J_1<0$ and $J_2=0$, using a contour
argument which we also develop in the paper. By the argument we also study
Gibbs measures for a natural generalization of the Ising model. We discuss some
open problems and state several conjectures.\
2829.
GIBBS MEASURES FOR SOS MODELS ON A CAYLEY TREE
U.A. Rozikov; Yu.M.Suhov
We consider a nearest-neighbor SOS model, spin values $0,1,..., m$, $m\geq
2$, on a Cayley tree of order $k$ . We mainly assume that $m=2$ and study
translation-invariant (TI) and `splitting' (S) Gibbs measures (GMs). For $m=2$,
in the anti-ferromagnetic (AFM) case, a symmetric TISGM is unique for all
temperatures. In the ferromagnetic (FM) case, for $m=2$, the number of
symmetric TISGMs varies with the temperature: here we identify a critical
inverse temperature, $\beta^1_{\rm{cr}}$ ($=T_{\rm{cr}}^{\rm{STISG}}$) $\in
(0,\infty)$ such that $\forall$ $0\leq \beta\leq\beta^1_{\rm{cr}}$, there
exists a unique symmetric TISGM $\mu^*$ and $\forall$ $\beta
>\beta^1_{\rm{cr}}$ there are exactly three symmetric TISGMs : $\mu^*_+$,
$\mu^*_{\rm m}$ and $\mu^*_-$ For $\beta>\beta^1_{\rm{cr}}$ we also construct a
continuum of distinct, symmertric SGMs which are non-TI. Our second result
gives complete description of the set of periodic Gibbs measures for the SOS
model on a Cayley tree. We show that (i) for an FM SOS model, for any normal
subgroup of finite index, each periodic SGM is in fact TI. Further, (ii) for an
AFM SOS model, for any normal subgroup of finite index, each periodic SGM is
either TI or has period two (i.e., is a chess-board SGM).
2830.
MALLIAVIN CALCULUS AND ERGODIC PROPERTIES OF HIGHLY DEGENERATE 2D
STOCHASTIC NAVIER--STOKES EQUATION
Martin Hairer, Jonathan C. Mattingly, Etienne Pardoux
The objective of this note is to present the results from the two recent
papers. We study the Navier--Stokes equation on the two--dimensional torus when
forced by a finite dimensional white Gaussian noise. We give conditions under
which both the law of the solution at any time t>0, projected on a finite
dimensional subspace, has a smooth density with respect to Lebesgue measure and
the solution itself is ergodic. In particular, our results hold for specific
choices of four dimensional white Gaussian noise. Under additional assumptions,
we show that the preceding density is everywhere strictly positive.
2831.
UNCERTAINTY RELATIONS IN MODELS OF MARKET MICROSTRUCTURE
Ted Theodosopoulos
This paper presents a new interacting particle system and uses it as a spin
model for financial market microstructure. The asymptotic analysis of this
stochastic process exhibits a lower bound to the contemporaneous measurement of
price and trading volume under the invariant measure in the `frozen' phase of
the supercritical regime.
2832.
TRANSLATION INVARIANT GIBBS STATES FOR THE ISING MODEL
T. Bodineau
We prove that all the translation invariant Gibbs states of the Ising model
are a linear combination of the pure phases $\mu^+,\mu^-$ in the phase
transition regime. This implies that the average magnetization is continuous in
the phase transition regime ($\beta > \beta_c$). Furthermore, combined with
previous results on the slab percolation threshold this shows the validity of
Pisztora's coarse graining up to the critical temperature.
2833.
NORMAL APPROXIMATION IN GEOMETRIC PROBABILITY
Mathew D. Penrose, J. E. Yukich
We use Stein's method to obtain bounds on the rate of convergence for a class
of statistics in geometric probability obtained as a sum of contributions from
Poisson points which are exponentially stabilizing, i.e. locally determined in
a certain sense. Examples include statistics such as total edge length and
total number of edges of graphs in computational geometry and the total number
of particles accepted in random sequential packing models. These rates also
apply to the 1-dimensional marginals of the random measures associated with
these statistics.
2834.
ON THE MARKOV CHAIN CENTRAL LIMIT THEOREM
Galin L. Jones
The goal of this mainly expository paper is to describe conditions which
guarantee a central limit theorem for functionals of general state space Markov
chains with a view towards Markov chain Monte Carlo settings. Thus the focus is
on the connections between drift and mixing conditions and their implications.
In particular, we consider three commonly cited central limit theorems and
discuss their relationship to classical results for mixing processes. Several
motivating examples are given which range from toy one-dimensional settings to
complicated settings encountered in Markov chain Monte Carlo.
2835.
ON THE BROWNIAN DIRECTED POLYMER IN A GAUSSIAN RANDOM ENVIRONMENT
Carles Rovira and amy Tindel
In this paper, we introduce a model of Brownian polymer in a continuous
random environment. The asymptotic behavior of the partition function
associated to this polymer measure is studied, and we are able to separate a
weak and strong disorder regime under some reasonable assumptions on the
spatial covariance of the environment. Some further developments, concerning
some concentration inequalities for the partition function, are given for the
weak disorder regime.
2836.
CHERNOFF'S THEOREM AND DISCRETE TIME APPROXIMATIONS OF BROWNIAN MOTION
ON MANIFOLDS
O. G. Somlyanov, H. v Weizsaecker and O. Wittich
Given a one-parameter family S = (S(t)) of bounded operators on a Banach
space whose derivative at time 0 is given by the closed operator A we prove a
version of Chernoffs Theorem which allows to approximate the semigroup
generated by A via compositions of members of S, indexed by nonuniform
partitions of the time axis. If A is an elliptic operator and the S(t) are
positive integral operators we prove tightness of the associated interpolating
path space measures. We apply this to the case where the underlying state space
is a submanifold L of a Riemannian manifold M and the S(t) are e.g. given by
the normalized restrictions of the heat kernel on M to L. In this case A is the
Laplace-Beltrami operator of L. Without normalization the limit measure has a
nontrivial Radon-Nikodym-density involving curvature terms with respect to
Wiener measure. These results substantially extend earlier work by the authors
and by Andersson and Driver.
2837.
ON RECENT PROGRESS FOR THE STOCHASTIC NAVIER STOKES EQUATIONS
Jonathan C. Mattingly
We give an overview of the ideas central to some recent developments in the
ergodic theory of the stochastically forced Navier Stokes equations and other
dissipative stochastic partial differential equations. Since our desire is to
make the core ideas clear, we will mostly work with a specific example: the
stochastically forced Navier Stokes equations. To further clarify ideas, we
will also examine in detail a toy problem. A few general theorems are given.
Spatial regularity, ergodicity, exponential mixing, coupling for a SPDE, and
hypoellipticity are all discussed.
2838.
ON THE TOTAL LENGTH OF THE RANDOM MINIMAL DIRECTED SPANNING TREE
Mathew D. Penrose and Andrew R. Wade
In Bhatt and Roy's minimal directed spanning tree (MDST) construction for a
random partially ordered set of points in the unit square,all edges must
respect the ``coordinatewise'' partial order and there must be a directed path
from each vertex to a minimal element. We study the asymptotic behaviour of the
total length of this graph with power weighted edges. The limiting distribution
is given by the sum of a normal component away from the boundary and a
contribution introduced by the boundary effects, which can be characterized by
a fixed point equation, and is reminiscent of limits arising in the
probabilistic analysis of certain algorithms. As the exponent of the power
weighting increases, the distribution undergoes a phase transition from the
normal contribution being dominant to the boundary effects dominating. In the
critical case where the weight is simple Euclidean length, both effects
contribute significantly to the limit law. We also give a law of large numbers
for the total weight of the graph.
2839.
ON LONG RANGE PERCOLATION WITH HEAVY TAILS
S. Friedli, N.B.N. de Lima, V. Sidoravicius
Consider independent long range percolation on $\mathbf{Z}^2$, where
horizontal and vertical edges of length $n$ are open with probability $p_n$. We
show that if $\limsup_{n\to\infty}p_n>0,$ then there exists an integer $N$ such
that $P_N(0\leftrightarrow \infty)>0$, where $P_N$ is the truncated measure
obtained by taking $p_{N,n}=p_n$ for $n \leq N$ and $p_{N,n}=0$ for all $n> N$.
2840.
AN LIL FOR COVER TIMES OF DISKS BY PLANAR RANDOM WALK AND WIENER SAUSAGE
J. Ben Hough, Yuval Peres
Let R_n be the radius of the largest disk covered after n steps of a simple
random walk. We prove that almost surely limsup_{n \to \infty}(log R_n)^2/(log
n log_3 n) = 1/4, where log_3 denotes 3 iterations of the log function. This is
motivated by a question of Erd\H{o}s and Taylor. We also obtain the analogous
result for the Wiener sausage, refining a result of Meyre and Werner.
2841.
FIRST EXIT TIMES OF SOLUTIONS OF NON-LINEAR STOCHASTIC DIFFERENTIAL
EQUATIONS DRIVEN BY SYMMETRIC LEVY PROCESSES WITH ALPHA-STABLE COMPONENTS
Peter Imkeller and Ilya Pavlyukevich
We study the exit problem of solutions of the stochastic differential
equation dX(t)=-U'(X(t))dt+epsilon dL(t) from bounded or unbounded intervals
which contain the unique asymptotically stable critical point of the
deterministic dynamical system dY=-U'(Y) dt. The process L is composed of a
standard Brownian motion and a symmetric alpha-stable Levy process. Using
probabilistic estimates we show that in the small noise limit epsilon->0, the
exit time of X from an interval is an exponentially distributed random variable
and determine its expected value. Due to the heavy-tail nature of the
alpha-stable component of L, the results differ strongly from the well known
case in which the deterministic dynamical system undergoes purely Gaussian
perturbations.
2842.
SOME REMARKS ABOUT THE POSITIVITY OF RANDOM VARIABLES ON A GAUSSIAN
PROBABILITY SPACE
D. Feyel and A.S. Ustunel
Let $(W,H,\mu)$ be an abstract Wiener space and $L$ be a probability density
of class LlogL. Using the measure transportation of Monge-Kantorovitch, we
prove that the kernel of the projection of L on the second Wiener chaos defines
an (Hilbert-Schmidt) operator which is lower bounded by another Hilbert-Schmidt
operator.
2843.
CUGLIANDOLO-KURCHAN EQUATIONS FOR DYNAMICS OF SPIN-GLASSES
Gerard Ben Arous, Amir Dembo, Alice Guionnet
We study the Langevin dynamics for the family of spherical $p$-spin
disordered mean-field models and prove that in the limit of system size $N$
approaching infinity, the empirical state correlation and integrated response
functions converge almost surely and uniformly in time, to the non-random
unique strong solution of a pair of explicit non-linear integro-differential
equations introduced by Cugliandolo and Kurchan.
2844.
LONG TIME BEHAVIOR OF THE SOLUTIONS TO NON-LINEAR KRAICHNAN EQUATIONS
Alice Guionnet, Christian Mazza
We consider the solution of a nonlinear Kraichnan equation $$\partial_s
H(s,t)=\int_t^s H(s,u)H(u,t) k(s,u) du,\quad s\ge t$$ with a covariance kernel
$k$ and boundary condition $H(t,t)=1$. We study the long time behaviour of $H$
as the time parameters $t,s$ go to infinity, according to the asymptotic
behaviour of $k$. This question appears in various subjects since it is related
with the analysis of the asymptotic behaviour of the trace of non-commutative
processes satisfying a linear differential equation, but also naturally shows
up in the study of the so-called response function and aging properties of the
dynamics of some disordered spin systems.
2845.
LARGE DEVIATIONS AND STOCHASTIC CALCULUS FOR LARGE RANDOM MATRICES
Alice Guionnet
These are notes from the lectures I gave at the XXIX conference on Stochastic
processes and Applications in August 2003. They intend to give a self-content
description of the use of large deviations techniques to study large random
matrices, and in particular (several) matrix models and Voiculescu's free
entropies, following a serie of papers I wrote with different coauthors. They
provide introduction to these topics and should be accessible to non (free)
probabilists.
2846.
RANDOM WALK LOOP SOUP
Gregory F. Lawler, Jos\'e A. Trujillo Ferreras
The Brownian loop soup introduced in Lawler and Werner (2004) is a Poissonian
realization from a sigma-finite measure on unrooted loops. This measure
satisfies both conformal invariance and a restriction property. In this paper,
we define a random walk loop soup and show that it converges to the Brownian
loop soup. In fact, we give a strong approximation result making use of the
strong approximation result of Koml\'os, Major, and Tusn\'ady. To make the
paper self-contained, we include a proof of the approximation result that we
need.
2847.
HARNESS PROCESSES AND NON-HOMOGENEOUS CRYSTALS
Pablo A. Ferrari, Beat M. Niederhauser, Eugene A. Pechersky
We consider the Harmonic crystal, a measure on R^{Z^d} with Hamiltonian H(x)
= \sum_{i,j} J_{i,j} (x(i)-x(j))^2 + h \sum_i (x(i)-d(i))^2, where x,d are
configurations. The configuration d is given and considered as observations.
The `couplings' J_{i,j} are finite range. We use a version of the harness
process to explicitly construct the infinite volume measure at finite
temperature and to find the unique configuration x' minimizing the Hamiltonian.
2848.
LARGE DEVIATIONS FOR THE CHEMICAL DISTANCE IN SUPERCRITICAL BERNOULLI
PERCOLATION
Olivier Garet, Regine Marchand
The chemical distance $D(x,y)$ is the length of the shortest open path
between two points $x$ and $y$ in an infinite Bernoulli percolation cluster. In
this work, we study the asymptotic behaviour of this random metric, and we
prove that, for an appropriate norm $\mu$ depending on the dimension and the
percolation parameter, the probability of the event $$\Big{0 \communique x,
\frac{D(0,x)}{\mu(x)}\notin (1-\epsilon, 1+\epsilon) \Big\}$$ exponentially
decreases when $\|x\|_1$ tends to infinity. From this bound we also derive a
large deviation inequality for the corresponding asymptotic shape result.
2849.
ONE-DIMENSIONAL LINEAR RECURSIONS WITH MARKOV-DEPENDENT COEFFICIENTS
Alexander Roitershtein
For a class of stationary Markov-dependent sequences (\xi_n,\rho_n) in \rr^2,
we consider the random linear recursion S_n=\xi_n+\rho_n S_{n-1} and show that
the distribution tail of its stationary solution has a power law decay. An
application to random walks in random environments is discussed.
2850.
CONCENTRATION OF THE BROWNIAN BRIDGE ON CARTAN-HADAMARD MANIFOLDS WITH
PINCHED NEGATIVE SECTIONAL CURVATURE
Marc Arnaudon, Thomas Simon
We study the rate of concentration of a Brownian bridge in time one around
the corresponding geodesical segment on a Cartan-Hadamard manifold with pinched
negative sectional curvature, when the distance between the two extremities
tends to infinity. This improves on previous results by A. Eberle, and one of
us. Along the way, we derive a new asymptotic estimate for the logarithmic
derivative of the heat kernel on such manifolds, in bounded time and with one
space parameter tending to infinity, which can be viewed as a counterpart to
Bismut's asymptotic formula in small time.
2851.
LARGE DEVIATIONS FOR PROCESSES WITH DISCONTINUOUS STATISTICS
Irina Ignatiouk-Robert
This paper is devoted to the problem of sample path large deviations for the
Markov processes on $\R_+^N$ describing a general class of queueing networks.
The global sample path large deviation principle and an integral representation
of the rate function are derived from local large deviation estimates. Our
results complete the proof of Dupuis and Ellis of the sample path large
deviation upper bound.
2852.
BOUNDING FASTEST MIXING
S. Roch
In a series of recent works, Boyd, Diaconis, and their co-authors have
introduced a semidefinite programming approach for computing the fastest mixing
Markov chain on a graph of allowed transitions, given a target stationary
distribution. In this paper, we show that standard mixing-time analysis
techniques--variational characterizations, conductance, canonical paths--can be
used to give simple, nontrivial lower and upper bounds on the fastest mixing
time. To test the applicability of this idea, we consider several detailed
examples including the Glauber dynamics of the Ising model--and get sharp
bounds.
2853.
REGULARITY OF THE SAMPLE PATHS OF A CLASS OF SECOND ORDER SPDE'S
Robert C. Dalang and Marta Sanz-Sol\'e
We study the sample path regularity of the solutions of a class of spde's
which are second order in time and that includes the stochastic wave equation.
Non-integer powers of the spatial Laplacian are allowed. The driving noise is
white in time and spatially homogeneous. Continuing with the work initiated in
Dalang and Mueller (2003), we prove that the solutions belong to a fractional
$L^2$-Sobolev space. We also prove H\"older continuity in time and therefore,
we obtain joint H\"older continuity in the time and space variables. Our
conclusions rely on a precise analysis of the properties of the stochastic
integral used in the rigourous formulation of the spde, as introduced by Dalang
and Mueller. For spatial covariances given by Riesz kernels, we show that our
results are optimal.
2854.
THE CLARK-OCONE FORMULA FOR VECTOR VALUED RANDOM VARIABLES IN ABSTRACT
WIENER SPACE
E. Mayer-Wolf and M. Zakai
The classical representation of random variables as the Ito integral of
nonanticipative integrands is extended to include Banach space valued random
variables on an abstract Wiener space equipped with a filtration induced by a
resolution of the identity on the Cameron-Martin space.
The Ito integral is replaced in this case by an extension of the divergence
to random operators, and the operators involved in the representation are
adapted with respect to this filtration in a suitably defined sense.
2855.
EXCEPTIONAL TIMES AND INVARIANCE FOR DYNAMICAL RANDOM WALKS
Davar Khoshnevisan, David A. Levin and Pedro J. Mendez-Hernandez
Consider a sequence {X(i,0) : i = 1, ..., n} of i.i.d. random variables.
Associate to each X(i,0) an independent mean-one Poisson clock. Every time a
clock rings replace that X-variable by an independent copy. In this way, we
obtain i.i.d. stationary processes {X(i,t) : t >= 0} (i=1,2, ...) whose
invariant distribution is the law of X(1,0). Benjamini, Haggstrom, Peres, and
Steif (2003) introduced the dynamical walk S(n,t) = X(1,t) + ... + X(n,t), and
proved among other things that the LIL holds for {S(n,t) : n =1,2, ...}
simultaneously for all t. In other words, the LIL is dynamically stable.
Subsequently, we showed that in the case that the X(i,0)'s are standard normal,
the classical integral test is not dynamically stable. Presently, we study the
set of times t when {S(n,t) : n=1,2, ...} exceeds a given envelope infinitely
often. Our analysis is made possible thanks to a connection to the Kolmogorov
epsilon-entropy. When used in conjunction with the invariance principle of this
paper, this connection has other interesting by-products some of which we
relate. We prove also that viewed as an infinite-dimensional process, the
rescaled dynamical random walk converges weakly in D(D([0,1])) to the
Ornstein-Uhlenbeck process in C([0,1]). For this we assume only that the
increments have mean zero and variance one. In addition, we extend a result of
Benjamini, Haggstrom, Peres and Steif (2003) by proving that if the X(i,0)'s
are lattice, mean-zero variance-one, and possess 2 + epsilon finite absolute
moments for some positive epsilon, then the recurrence of the origin is
dynamically stable. To prove this we derive a gambler's ruin estimate that is
valid for all lattice random walks that have mean zero and finite variance. We
believe the latter may be of independent interest.
2856.
IMAGES OF THE BROWNIAN SHEET
Davar Khoshnevisan and Yimin Xiao
An N-parameter Brownian sheet in R^d maps a non-random compact set F in R^N_+
to the random compact set B(F) in \R^d. We prove two results on the image-set
B(F):
(1) It has positive d-dimensional Lebesgue measure if and only if F has
positive (d/2)-dimensional capacity. This generalizes greatly the earlier works
of J. Hawkes (1977), J.-P. Kahane (1985a; 1985b), and one of the present
authors (1999).
(2) If the Hausdorff dimension of F is strictly greater than (d/2), then with
probability one, we can find a finite number of points \zeta_1,...,\zeta_m such
that for any rotation matrix \theta that leaves F in B(\theta F), one of the
\zeta_i's is interior to B(\theta F). In particular, B(F) has interior-points
a.s. This verifies a conjecture of T. S. Mountford (1989).
This paper contains two novel ideas: To prove (1), we introduce and analyze a
family of bridged sheets. Item (2) is proved by developing a notion of
``sectorial local-non-determinism (LND).'' Both ideas may be of independent
interest.
We showcase sectorial LND further by exhibiting some arithmetic properties of
standard Brownian motion; this completes the work initiated by Mountford
(1988).
2857.
LARGE TIME ASYMPTOTICS FOR THE DENSITY OF A BRANCHING WIENER PROCESS
P\'{a}l R\'{e}v\'{e}sz, Jay Rosen, Zhan Shi
Given an R^d-valued supercritical branching Wiener process, let D(A,T) be the
number of particles in a subset A of R^d at time T, (T=0,1,2,...). We provide a
complete asymptotic expansion of D(A,T) as T goes to infinity, generalizing the
work of X.Chen.
2858.
CONSTRUCTION OF A SPECIFICATION FROM ITS SINGLETON PART
Roberto Fernandez, Gregory Maillard
We state a construction theorem for specifications starting from single-site
conditional probabilities (singleton part). The result holds for general
single-site spaces under weak non-nullness requirements on the singleton part.
2859.
DISCRETIZATION METHODS FOR HOMOGENEOUS FRAGMENTATIONS
Jean Bertoin, Alain Rouault
Homogeneous fragmentations describe the evolution of a unit mass that breaks
down randomly into pieces as time passes. They can be thought of as continuous
time analogs of a certain type of branching random walks, which suggests the
use of time-discretization to shift known results from the theory of branching
random walks to the fragmentation setting. In particular, this yields
interesting information about the asymptotic behaviour of fragmentations.
On the other hand, homogeneous fragmentations can also be investigated using
a powerful technique of discretization of space due to Kingman, namely, the
theory of exchangeable partitions of $\N$. Spatial discretization is especially
well-suited to develop directly for continuous times the conceptual method of
probability tilting of Lyons, Pemantle and Peres.
2860.
ASYMPTOTICAL BEHAVIOUR OF THE PRESENCE PROBABILITY IN BRANCHING RANDOM
WALKS AND FRAGMENTATIONS
Jean Bertoin, Alain Rouault
For a subcritical Galton-Watson process $(\zeta_n)$, it is well known that
under an $X \log X$ condition, the quotient $P(\zeta_n > 0)/ E\zeta_n$ has a
finite positive limit. There is an analogous result for a (one-dimensional)
supercritical branching random walk: when $a$ is in the so-called subcritical
speed area, the probability of presence around $na$ in the $n$-th generation is
asymptotically proportional to the corresponding expectation. In Rouault (1993)
this result was stated under a natural $X \log X$ assumption on the offspring
point process and a (unnatural) condition on the offspring mean. Here we prove
that the result holds without this latter condition, in particular we allow an
infinite mean and a dimension $d \geq 1$ for the state-space. As a consequence
the result holds also for homogeneous fragmentations as defined in Bertoin
(2001), using the method of discrete-time skeletons; this completes the proof
of Theorem 4 in Bertoin-Rouault (2004 see math/PR/0409545). Finally, an
application to conditioning on the presence allows to meet again the
probability tilting and the so-called additive martingale.
2861.
ON MUTUAL INFORMATION, LIKELIHOOD-RATIOS AND ESTIMATION ERROR FOR THE
ADDITIVE GAUSSIAN CHANNEL
Moshe Zakai
This paper considers the model of an arbitrary distributed signal x observed
through an added independent white Gaussian noise w, y=x+w. New relations
between the minimal mean square error of the non-causal estimator and the
likelihood ratio between y and \omega are derived. This is followed by an
extended version of a recently derived relation between the mutual information
I(x;y) and the minimal mean square error. These results are applied to derive
infinite dimensional versions of the Fisher information and the de Bruijn
identity. The derivation of the results is based on the Malliavin calculus.
2862.
LARGE DEVIATION FOR THE EMPIRICAL EIGENVALUE DENSITY OF TRUNCATED HAAR
UNITARY MATRICES
Denes Petz, Julia Reffy
Let $U_m$ be an $m \times m$ Haar unitary matrix and $U_{[m,n]}$ be its $n
\times n$ truncation. In this paper the large deviation is proven for the
empirical eigenvalue density of $U_{[m,n]}$ as $m/n \to \lambda $ and $n \to
\infty$. The rate function and the limit distribution are given explicitely.
$U_{[m,n]}$ is the random matrix model of $quq$, where $u$ is a Haar unitary in
a finite von Neumann algebra, $q$ is a certain projection and they are free.
The limit distribution coincides with the Brown measure of the operator $quq$.
2863.
EXPECTATIONS OF HOOK PRODUCTS ON LARGE PARTITIONS
Mark Adler, Alexei Borodin and Pierre van Moerbeke
Given uniform probability on words of length M=Np+k, from an alphabet of size
p, consider the probability that a word
(i) contains a subsequence of letters (p, p-1,...,1) in that order and
(ii) that the maximal length of the disjoint union of p-1 increasing
subsequences of the word is \leq M-N . A generating function for this
probability has the form of an integral over the Grassmannian of p-planes in
complex C^n. The present paper shows that the asymptotics of this probability,
when N tends to infinity, is related to the kth moment of the
chi^2-distribution of parameter 2p^2. This is related to the behavior of the
integral over the Grassmannian Gr(p,C^n) of p-planes in C^n, when the dimension
of the ambient space C^n becomes very large. A different scaling limit for the
Poissonized probability is related to a new matrix integral, itself a solution
of the Painlev\'e IV equation. This is part of a more general set-up related to
the Painlev\'e V equation.
2864.
SYMMETRIC RANDOM WALKS ON CERTAIN AMALGAMATED FREE PRODUCT GROUPS
Ken Dykema
We consider nearest-neighbor random walks on free products of finitely many
copies of the integers with amalgamation over nontrivial subgroups. When all
the subgroups have index two, we find the Green function of the random walks in
terms of complete elliptic integrals. We prove that in all cases, the spectral
radius of the random walk is an algebraic number and we show how to find its
value, doing so in a few cases. Our technique is to apply Voiculescu's
operator--valued R--transform.
2865.
AN ASYMPTOTIC BERRY-ESSEEN RESULT FOR THE LARGEST EIGENVALUE OF COMPLEX
WHITE WISHART MATRICES
Noureddine El Karoui
A number of results concerning the convergence in distribution of the largest
eigenvalue of a large class of random covariance matrices have recently been
obtained. In particular, it was shown in by Johansson (2000), Johnstone (2001),
and El Karoui (2003) that if X is an n*N matrix whose entries are i.i.d
standard complex Gaussian and l_1 is the largest eigenvalue of X^*X, there
exist sequences m_{n,N} and s_{n,N} such that (l_1-m_{n,N})/s_{n,N} converges
in distribution to the Tracy-Widom law of order 2, denoted W_2, a distribution
whose density is known and computable. Its cumulative distribution function is
denoted F_2.
In this paper, we show that we can find a function M, and sequences mu_{n,N}
and sigma_{n,N} such that when n and N go to infinity, and n/N tends to gamma
(a non-zero real number), we have, with l_{n,N}=(l_1-mu_{n,N})/sigma_{n,N}, for
all s in R, min(n,N)^{2/3} |P(l_{n,N}<=s)-F_2(s)|<= M(s) . The surprisingly
good 2/3 rate helps explain the fact that the limiting distribution F_2 is a
good approximation to the empirical distribution of l_{n,N} in simulations, an
important fact from the point of view of (for instance, statistical)
applications.
2866.
REFLECTED BROWNIAN MOTION IN GENERIC TRIANGLES AND WEDGES
Wouter Kager
Consider a generic triangle in the upper half of the complex plane with one
side on the real line. We construct a random walk whose scaling limit is a
Brownian motion in the triangle, reflected on the left and right boundaries
with constant reflection angles. The construction is such that the exit point
of the reflected Brownian motion from the triangle is uniformly distributed on
the base. This generalizes earlier work of Julien Dubedat on reflected Brownian
motion in isosceles triangles. We also compute several distribution functions
associated with the reflected Brownian motions.
2867.
GEOMETRIC ERGODICITY AND PERFECT SIMULATION
Wilfrid S. Kendall
This note extends the work of Foss and Tweedie (1997), who showed that
availability of the classic Coupling from The Past algorithm of Propp and
Wilson (1996) is essentially equivalent to uniform ergodicity for a Markov
chain (see also HobertRobert, 2004). In this note we show that all
geometrically ergodic chains possess dominated Coupling from The Past
algorithms (not necessarily practical!) which are rather closely connected to
Foster-Lyapunov criteria.
2868.
MULTIVARIATE SPATIAL CENTRAL LIMIT THEOREMS WITH APPLICATIONS TO
PERCOLATION AND SPATIAL GRAPHS
Mathew D Penrose
Suppose $X = (X_x, x$ in $Z^d)$ is a family of i.i.d. variables in some
measurable space, $B_0$ is a bounded set in $R^d$, and for $t > 1$, $H_t$ is a
measure on $tB_0$ determined by the restriction of $X$ to lattice sites in or
adjacent to $tB_0$. We prove convergence to a white noise process for the
random measure on $B_0$ given by $t^{-d/2}(H_t(tA)-EH_t(tA))$ for subsets $A$
of $B_0$, as $t$ becomes large,subject to $H$ satisfying a ``stabilization''
condition (whereby the effect of changing $X$ at a single site $x$ is local)
but with no assumptions on the rate of decay of correlations. We also give a
multivariate central limit theorem for the joint distributions of two or more
such measures $H_t$, and adapt the result to measures based on Poisson and
binomial point processes. Applications given include a white noise limit for
the measure which counts clusters of critical percolation, a functional central
limit theorem for the empirical process of the edge lengths of the minimal
spanning tree on random points, and central limit theorems for the on-line
nearest neighbour graph.
2869.
A UNIVERSALITY PROPERTY FOR LAST-PASSAGE PERCOLATION PATHS CLOSE TO THE
AXIS
Thierry Bodineau and James B. Martin
We consider a last-passage directed percolation model in $Z_+^2$, with i.i.d.
weights whose common distribution has a finite $(2+p)$th moment. We study the
fluctuations of the passage time from the origin to the point
$\big(n,n^{\lfloor a \rfloor}\big)$. We show that, for suitable $a$ (depending
on $p$), this quantity, appropriately scaled, converges in distribution as
$n\to\infty$ to the Tracy-Widom distribution, irrespective of the underlying
weight distribution. The argument uses a coupling to a Brownian directed
percolation problem and the strong approximation of Koml\'os, Major and
Tusn\'ady.
2870.
PARAMETRIZED KANTOROVICH-RUBINSTEIN THEOREM AND APPLICATION TO THE
COUPLING OF RANDOM VARIABLES
Jerome Dedecker, Clementine Prieur, Paul Raynaud De
Fitte
We prove a version for random measures of the celebrated
Kantorovich-Rubinstein duality theorem and we give an application to the
coupling of random variables which extends and unifies known results.
2871.
VALIDITY OF HEAVY TRAFFIC STEADY-STATE APPROXIMATIONS IN OPEN QUEUEING
NETWORKS
David Gamarnik and Assaf Zeevi
We consider a single class open queueing network, also known as a Generalized
Jackson Network (GJN). A classical result in heavy traffic theory asserts that
the sequence of normalized queue length processes of the GJN converge weakly to
a Reflected Brownian Motion (RBM) in the orthant, as the traffic intensity
approaches unity. However, barring simple instances, it is still not known
whether the stationary distribution of RBM provides a valid approximation for
the steady-state of the original network.
In this paper we resolve this open problem by proving that the re-scaled
stationary distribution of the GJN converges to the stationary distribution of
the RBM, thus validating a so-called ``interchange-of-limits'' for this class
of networks. Our method of proof involves a combination of Lyapunov function
techniques, strong approximations, and tail probability bounds that yield
tightness of the sequence of stationary distributions of the GJN.
2872.
THE SHATTERING DIMENSION OF SETS OF LINEAR FUNCTIONALS
Shahar Mendelson and Gideon Schechtman
We evaluate the shattering dimension of various classes of linear functionals
on various symmetric convex sets. The proofs here relay mostly on methods from
the local theory of normed spaces and include volume estimates, factorization
techniques and tail estimates of norms, viewed as random variables on Euclidean
spheres. The estimates of shattering dimensions can be applied to obtain error
bounds for certain classes of functions, a fact which was the original
motivation of this study. Although this can probably be done in a more
traditional manner, we also use the approach presented here to determine
whether several classes of linear functionals satisfy the uniform law of large
numbers and the uniform central limit theorem.
2873.
CONVERGENCE OF FUNCTIONALS OF SUMS OF R.V.S TO LOCAL TIMES OF FRACTIONAL
STABLE MOTIONS
P. Jeganathan
Consider a sequence X_k=\sum_{j=0}^{\infty}c_j\xi_{k-j}, k\geq 1, where c_j,
j\geq 0, is a sequence of constants and \xi_j, -\infty <j<\infty, is a sequence
of independent identically distributed (i.i.d.) random variables (r.v.s)
belonging to the domain of attraction of a strictly stable law with index
0<\alpha \leq 2. Let S_k=\sum_{j=1}^kX_j. Under suitable conditions on the
constants c_j it is known that for a suitable normalizing constant \gamma_n,
the partial sum process \gamma_n^{-1}S_{[nt]} converges in distribution to a
linear fractional stable motion (indexed by \alpha and H, 0<H<1). A fractional
ARIMA process with possibly heavy tailed innovations is a special case of the
process X_k. In this paper it is established that the process
n^{-1}\beta_n\sum_{k=1}^{[nt]}f(\beta_n(\gamma_n^{-1}S_k+x)) converges in
distribution to (\int_{-\infty}^{\infty}f(y) dy)L(t,-x), where L(t,x) is the
local time of the linear fractional stable motion, for a wide class of
functions f(y) that includes the indicator functions of bounded intervals of
the real line. Here \beta_n\to \infty such that n^{-1}\beta_n\to 0. The only
further condition that is assumed on the distribution of \xi_1 is that either
it satisfies the Cram\'er's condition or has a nonzero absolutely continuous
component.
The results have motivation in large sample inference for certain nonlinear
time series models.
2874.
HOEFFDING-ANOVA DECOMPOSITIONS FOR SYMMETRIC STATISTICS OF EXCHANGEABLE
OBSERVATIONS
Giovanni Peccati
Consider a (possibly infinite) exchangeable sequence X={X_n:1\leqn<N}, where
N\in N\cup {\infty}, with values in a Borel space (A,A), and note
X_n=(X_1,...,X_n). We say that X is Hoeffding decomposable if, for each n,
every square integrable, centered and symmetric statistic based on X_n can be
written as an orthogonal sum of n U-statistics with degenerated and symmetric
kernels of increasing order. The only two examples of Hoeffding decomposable
sequences studied in the literature are i.i.d. random variables and extractions
without replacement from a finite population. In the first part of the paper we
establish a necessary and sufficient condition for an exchangeable sequence to
be Hoeffding decomposable, that is, called weak independence. We show that not
every exchangeable sequence is weakly independent, and, therefore, that not
every exchangeable sequence is Hoeffding decomposable. In the second part we
apply our results to a class of exchangeable and weakly independent random
vectors X_n^{(\alpha, c)}=(X_1^{(\alpha, c)},...,X_n^{(\alpha, c)}) whose law
is characterized by a positive and finite measure \alpha (\cdot) on A and by a
real constant c. For instance, if c=0, X_n^{(\alpha, c)} is a vector of i.i.d.
random variables with law \alpha (\cdot)/\alpha (A); if A is finite, \alpha
(\cdot) is integer valued and c=-1, X_n^{(\alpha, c)} represents the first n
extractions without replacement from a finite population; if c>0, X_n^{(\alpha,
c)} consists of the first n instants of a generalized P\'olya urn sequence.
2875.
A GENERAL NONCONVEX LARGE DEVIATION RESULT II
A. de Acosta
We refine the conditions for the lower bound in an abstract large deviation
result with nonconvex rate function we had previously introduced. We apply the
results to certain stochastic recursive schemes.
2876.
SELF-NORMALIZED PROCESSES: EXPONENTIAL INEQUALITIES, MOMENT BOUNDS AND
ITERATED LOGARITHM LAWS
Victor H. de la Pena, Michael J. Klass and Tze Leung Lai
Self-normalized processes arise naturally in statistical applications.
Being unit free, they are not affected by scale changes. Moreover,
self-normalization often eliminates or weakens moment assumptions. In this
paper we present several exponential and moment inequalities, particularly
those related to laws of the iterated logarithm, for self-normalized random
variables including martingales. Tail probability bounds are also derived. For
random variables B_t>0 and A_t, let Y_t(\lambda)=\exp{\lambda A_t-\lambda
^2B_t^2/2}. We develop inequalities for the moments of A_t/B_{t} or sup_{t\geq
0}A_t/{B_t(\log \log B_{t})^{1/2}} and variants thereof, when EY_t(\lambda
)\leq 1 or when Y_t(\lambda) is a supermartingale, for all \lambda belonging to
some interval. Our results are valid for a wide class of random processes
including continuous martingales with A_t=M_t and B_t=\sqrt < M>_t, and sums of
conditionally symmetric variables d_i with A_t=\sum_{i=1}^td_i and
B_t=\sqrt\sum_{i=1}^td_i^2. A sharp maximal inequality for conditionally
symmetric random variables and for continuous local martingales with values in
R^m, m\ge 1, is also established. Another development in this paper is a
bounded law of the iterated logarithm for general adapted sequences that are
centered at certain truncated conditional expectations and self-normalized by
the square root of the sum of squares. The key ingredient in this development
is a new exponential supermartingale involving \sum_{i=1}^td_i and
\sum_{i=1}^td_i^2.
2877.
CENTRAL LIMIT THEOREMS FOR ITERATED RANDOM LIPSCHITZ MAPPINGS
Hubert Hennion and Loic Herve
Let M be a noncompact metric space in which every closed ball is compact, and
let G be a semigroup of Lipschitz mappings of M. Denote by (Y_n)_{n\geq1} a
sequence of independent G-valued, identically distributed random variables
(r.v.'s), and by Z an M-valued r.v. which is independent of the r.v. Y_n,
n\geq1. We consider the Markov chain (Z_n)_{n\geq0} with state space M which is
defined recursively by Z_0=Z and Z_{n+1}=Y_{n+1}Z_n for n\geq0.
Let \xi be a real-valued function on G\times M. The aim of this paper is to
prove central limit theorems for the sequence of r.v.'s
(\xi(Y_n,Z_{n-1}))_{n\geq1}.
The main hypothesis is a condition of contraction in the mean for the action
on M of the mappings Y_n; we use a spectral method based on a quasi-compactness
property of the transition probability of the chain mentioned above, and on a
special perturbation theorem.
2878.
NORMAL APPROXIMATION UNDER LOCAL DEPENDENCE
Louis H. Y. Chen and Qi-Man Shao
We establish both uniform and nonuniform error bounds of the Berry-Esseen
type in normal approximation under local dependence. These results are of an
order close to the best possible if not best possible. They are more general or
sharper than many existing ones in the literature. The proofs couple Stein's
method with the concentration inequality approach.
2879.
LIMIT THEOREMS FOR A CLASS OF IDENTICALLY DISTRIBUTED RANDOM VARIABLES
Patrizia Berti, Luca Pratelli and Pietro Rigo
A new type of stochastic dependence for a sequence of random variables is
introduced and studied. Precisely, (X_n)_{n\geq 1} is said to be conditionally
identically distributed (c.i.d.), with respect to a filtration (G_n)_{n\geq 0},
if it is adapted to (G_n)_{n\geq 0} and, for each n\geq 0, (X_k)_{k>n} is
identically distributed given the past G_n. In case G_0={\varnothing,\Omega}
and G_n=\sigma(X_1,...,X_n), a result of Kallenberg implies that (X_n)_{n\geq
1} is exchangeable if and only if it is stationary and c.i.d. After giving some
natural examples of nonexchangeable c.i.d. sequences, it is shown that
(X_n)_{n\geq 1} is exchangeable if and only if (X_{\tau(n)})_{n\geq 1} is
c.i.d. for any finite permutation \tau of {1,2,...}, and that the distribution
of a c.i.d. sequence agrees with an exchangeable law on a certain
sub-\sigma-field. Moreover, (1/n)\sum_{k=1}^nX_k converges a.s. and in L^1
whenever (X_n)_{n\geq 1} is (real-valued) c.i.d. and E[|
X_1| ]<\infty. As to the CLT, three types of random centering are considered.
One such centering, significant in Bayesian prediction and discrete time
filtering, is E[X_{n+1}| G_n]. For each centering, convergence in distribution
of the corresponding empirical process is analyzed under uniform distance.
2880.
P-VARIATION OF STRONG MARKOV PROCESSES
Martynas Manstavicius
Let \xi_t, t\in[0,T], be a strong Markov process with values in a complete
separable metric space (X,\rho) and with transition probability function
P_{s,t}(x,dy), 0\le s\le t\le T, x\in X. For any h\in[0,T] and a>0, consider
the function \alpha(h,a)=sup\bigl{P_{s,t}\bigl(x,{y:\rho(x,y)\ge a}\bigr):x\in
X,0\le s\le t\le (s+h)\wedge T\bigr}. It is shown that a certain growth
condition on \alpha(h,a), as a\downarrow0 and h stays fixed, implies the almost
sure boundedness of the p-variation of \xi_t, where p depends on the rate of
growth.
2881.
ABSOLUTE CONTINUITY OF SYMMETRIC MARKOV PROCESSES
Z.-Q. Chen, P. J. Fitzsimmons, M. Takeda, J. Ying and T.-S. Zhang
We study Girsanov's theorem in the context of symmetric Markov processes,
extending earlier work of Fukushima-Takeda and Fitzsimmons on Girsanov
transformations of ``gradient type.'' We investigate the most general Girsanov
transformation leading to another symmetric Markov process. This investigation
requires an extension of the forward-backward martingale method of Lyons-Zheng,
to cover the case of processes with jumps.
2882.
POTENTIAL THEORY FOR HYPERBOLIC SPDES
Robert C. Dalang and Eulalia Nualart
We give general sufficient conditions which imply upper and lower bounds for
the probability that a multiparameter process hits a given set E in terms of a
capacity of E related to the process. This extends a result of Khoshnevisan and
Shi [Ann. Probab. 27 (1999) 1135-1159], where estimates for the hitting
probabilities of the (N,d) Brownian sheet in terms of the
(d-2N) Newtonian capacity are obtained, and readily applies to a wide class of
Gaussian processes. Using Malliavin calculus and, in particular, a result of
Kohatsu-Higa [Probab. Theory Related Fields 126 (2003) 421-457], we apply these
general results to the solution of a system of d nonlinear hyperbolic
stochastic partial differential equations with two variables.
We show that under standard hypotheses on the coefficients, the hitting
probabilities of this solution are bounded above and below by constants times
the (d-4) Newtonian capacity. As a consequence, we characterize polar sets for
this process and prove that the Hausdorff dimension of its range is min(d,4)
a.s.
2883.
CALCULATING THE GREEKS BY CUBATURE FORMULAS
Josef Teichmann
We provide cubature formulas for the calculation of derivatives of expected
values in the spririt of Terry Lyons and Nicolas Victoir. In financial
mathematics derivatives of option prices with respect to initial values, so
called Greeks, are of particular importance as hedging parameters. Cubature
formulas allow to calculate these quantities very quickly. Simple examples are
added to the theoretical exposition.
2884.
TRIMMED TREES AND EMBEDDED PARTICLE SYSTEMS
Klaus Fleischmann and Jan M. Swart
In a supercritical branching particle system, the trimmed tree consists of
those particles which have descendants at all times. We develop this concept in
the superprocess setting. For a class of continuous superprocesses with Feller
underlying motion on compact spaces, we identify the trimmed tree, which turns
out to be a binary splitting particle system with a new underlying motion that
is a compensated h-transform of the old one. We show how trimmed trees may be
estimated from above by embedded binary branching particle systems.
2885.
STABLE STATIONARY PROCESSES RELATED TO CYCLIC FLOWS
Vladas Pipiras and Murad S. Taqqu
We study stationary stable processes related to periodic and cyclic flows in
the sense of Rosinski [Ann. Probab. 23 (1995) 1163-1187]. These processes are
not ergodic. We provide their canonical representations, consider examples and
show how to identify them among general stationary stable processes.
We conclude with the unique decomposition in distribution of stationary stable
processes into the sum of four major independent components: 1.
A mixed moving average component. 2. A harmonizable (or ``trivial'')
component.
3. A cyclic component 4. A component which is different from these.
2886.
UTILITY MAXIMIZING ENTROPY AND THE SECOND LAW OF THERMODYNAMICS
Wojciech Slomczynski and Tomasz Zastawniak
Expected utility maximization problems in mathematical finance lead to a
generalization of the classical definition of entropy. It is demonstrated that
a necessary and sufficient condition for the second law of thermodynamics to
operate is that any one of the generalized entropies should tend to its minimum
value of zero.
2887.
CONDITIONING AND INITIAL ENLARGEMENT OF FILTRATION ON A RIEMANNIAN
MANIFOLD
Fabrice Baudoin
We extend to Riemannian manifolds the theory of conditioned stochastic
differential equations. We also provide some enlargement formulas for the
Brownian filtration in this nonflat setting.
2888.
THE EULER SCHEME FOR LEVY DRIVEN STOCHASTIC DIFFERENTIAL EQUATIONS:
LIMIT THEOREMS
Jean Jacod
We study the Euler scheme for a stochastic differential equation driven by a
Levy process Y. More precisely, we look at the asymptotic behavior of the
normalized error process u_n(X^n-X), where X is the true solution and X^n is
its Euler approximation with stepsize 1/n, and u_n is an appropriate rate going
to infinity: if the normalized error processes converge, or are at least tight,
we say that the sequence (u_n) is a rate, which, in addition, is sharp when the
limiting process (or processes) is not trivial.
We suppose that Y has no Gaussian part (otherwise a rate is known to be
u_n=\sqrt n). Then rates are given in terms of the concentration of the Levy
measure of Y around 0 and, further, we prove the convergence of the sequence
u_n(X^n-X) to a nontrivial limit under some further assumptions, which cover
all stable processes and a lot of other Levy processes whose Levy measure
behave like a stable Levy measure near the origin. For example, when Y is a
symmetric stable process with index \alpha \in(0,2), a sharp rate is
u_n=(n/\log n)^{1/\alpha}; when Y is stable but not symmetric, the rate is
again u_n=(n/\log n)^{1/\alpha} when \alpha >1, but it becomes u_n=n/(\log n)^2
if \alpha =1 and u_n=n if \alpha <1.
2889.
ZERO TEMPERATURE LIMIT FOR INTERACTING BROWNIAN PARTICLES. I. MOTION OF
A SINGLE BODY
Tadahisa Funaki
We consider a system of interacting Brownian particles in R^d with a pairwise
potential, which is radially symmetric, of finite range and attains a unique
minimum when the distance of two particles becomes a>0. The asymptotic behavior
of the system is studied under the zero temperature limit from both microscopic
and macroscopic aspects. If the system is rigidly crystallized, namely if the
particles are rigidly arranged in an equal distance a, the crystallization is
kept under the evolution in macroscopic time scale.
Then, assuming that the crystal has a definite limit shape under a macroscopic
spatial scaling, the translational and rotational motions of such shape are
characterized.
2890.
ZERO TEMPERATURE LIMIT FOR INTERACTING BROWNIAN PARTICLES. II.
COAGULATION IN ONE DIMENSION
Tadahisa Funaki
We study the zero temperature limit for interacting Brownian particles in one
dimension with a pairwise potential which is of finite range and attains a
unique minimum when the distance of two particles becomes a>0.
We say a chain is formed when the particles are arranged in an ``almost
equal'' distance a. If a chain is formed at time 0, so is for positive time as
the temperature of the system decreases to 0 and, under a suitable macroscopic
space-time scaling, the center of mass of the chain performs the Brownian
motion with the speed inversely proportional to the total mass. If there are
two chains, they independently move until the time when they meet. Then, they
immediately coalesce and continue the evolution as a single chain. This can be
extended for finitely many chains.
2891.
MODERATE DEVIATION PROBABILITIES FOR OPEN CONVEX SETS: NONLOGARITHMIC
BEHAVIOR
Uwe Einmahl and James Kuelbs
Precise asymptotics for moderate deviation probabilities are established for
open convex sets in both the finite- and infinite-dimensional settings.
Our results are based on the existence of dominating points for these sets, a
related representation formula, and asymptotics for the integral term in this
formula.
2892.
PATH DECOMPOSITIONS FOR MARKOV CHAINS
Gotz Kersting and Kaya Memisoglu
We present two path decompositions of Markov chains (with general state
space) by means of harmonic functions, which are dual to each other. They can
be seen as a generalization of Williams' decomposition of a Brownian motion
with drift. The results may be illustrated by a multitude of examples, but we
confine ourselves to different types of random walks and the Polya urn.
2893.
A UNIFORM FUNCTIONAL LAW OF THE LOGARITHM FOR THE LOCAL EMPIRICAL
PROCESS
David M. Mason
We prove a uniform functional law of the logarithm for the local empirical
process. To accomplish this we combine techniques from classical and abstract
empirical process theory, Gaussian distributional approximation and probability
on Banach spaces. The body of techniques we develop should prove useful to the
study of the strong consistency of d-variate kernel-type nonparametric function
estimators.
2894.
EXACT CONVERGENCE RATE AND LEADING TERM IN CENTRAL LIMIT THEOREM FOR
STUDENT'S T STATISTIC
Peter Hall and Qiying Wang
The leading term in the normal approximation to the distribution of Student's
t statistic is derived in a general setting, with the sole assumption being
that the sampled distribution is in the domain of attraction of a normal law.
The form of the leading term is shown to have its origin in the way in which
extreme data influence properties of the Studentized sum. The leading-term
approximation is used to give the exact rate of convergence in the central
limit theorem up to order n^{-1/2}, where n denotes sample size. It is proved
that the exact rate uniformly on the whole real line is identical to the exact
rate on sets of just three points. Moreover, the exact rate is identical to
that for the non-Studentized sum when the latter is normalized for scale using
a truncated form of variance, but when the corresponding truncated centering
constant is omitted. Examples of characterizations of convergence rates are
also given. It is shown that, in some instances, their validity uniformly on
the whole real line is equivalent to their validity on just two symmetric
points.
2895.
EXTREME VALUE THEORY, ERGODIC THEORY AND THE BOUNDARY BETWEEN SHORT
MEMORY AND LONG MEMORY FOR STATIONARY STABLE PROCESSES
Gennady Samorodnitsky
We study the partial maxima of stationary \alpha-stable processes. We relate
their asymptotic behavior to the ergodic theoretical properties of the flow. We
observe a sharp change in the asymptotic behavior of the sequence of partial
maxima as flow changes from being dissipative to being conservative, and argue
that this may indicate a change from a short memory process to a long memory
process.
2896.
MEANS OF A DIRICHLET PROCESS AND MULTIPLE HYPERGEOMETRIC FUNCTIONS
Antonio Lijoi and Eugenio Regazzini
The Lauricella theory of multiple hypergeometric functions is used to shed
some light on certain distributional properties of the mean of a Dirichlet
process. This approach leads to several results, which are illustrated here.
Among these are a new and more direct procedure for determining the exact form
of the distribution of the mean, a correspondence between the distribution of
the mean and the parameter of a Dirichlet process, a characterization of the
family of Cauchy distributions as the set of the fixed points of this
correspondence, and an extension of the Markov-Krein identity. Moreover, an
expression of the characteristic function of the mean of a Dirichlet process is
obtained by resorting to an integral representation of a confluent form of the
fourth Lauricella function. This expression is then employed to prove that the
distribution of the mean of a Dirichlet process is symmetric if and only if the
parameter of the process is symmetric, and to provide a new expression of the
moment generating function of the variance of a
Dirichlet process.
2897.
ON THE CONCENTRATION OF MEASURE PHENOMENON FOR STABLE AND RELATED RANDOM
VECTORS
Christian Houdre and Philippe Marchal
Concentration of measure is studied, and obtained, for stable and related
random vectors.
2898.
STOCHASTIC BOUNDS FOR LEVY PROCESSES
R. A. Doney
Using the Wiener-Hopf factorization, it is shown that it is possible to bound
the path of an arbitrary Levy process above and below by the paths of two
random walks. These walks have the same step distribution, but different random
starting points. In principle, this allows one to deduce Levy process versions
of many known results about the large-time behavior of random walks. This is
illustrated by establishing a comprehensive theorem about Levy processes which
converge to \infty in probability.
2899.
LEVY PROCESSES AND FOURIER ANALYSIS ON COMPACT LIE GROUPS
Ming Liao
We study the Fourier expansion of the distribution density of a Levy process
in a compact Lie group based on the Peter-Weyl theorem.
2900.
ALGEBRAIC METHODS TOWARD HIGHER-ORDER PROBABILITY INEQUALITIES, II
Donald St. P. Richards
Let (L,\preccurlyeq) be a finite distributive lattice, and suppose that the
functions f_1,f_2:L\to R are monotone increasing with respect to the partial
order \preccurlyeq. Given \mu a probability measure on L, denote by E(f_i) the
average of f_i over L with respect to \mu, i=1,2. Then the
FKG inequality provides a condition on the measure \mu under which the
covariance, Cov(f_1,f_2):=E(f_1f_2)-E(f_1)E(f_2), is nonnegative. In this paper
we derive a ``third-order'' generalization of the FKG inequality.
We also establish fourth- and fifth-order generalizations of the FKG
inequality and formulate a conjecture for a general mth-order generalization.
For functions and measures on R^n we establish these inequalities by extending
the method of diffusion processes. We provide several applications of the
third-order inequality, generalizing earlier applications of the FKG
inequality.
Finally, we remark on some connections between the theory of total positivity
and the existence of inequalities of FKG-type within the context of Riemannian
manifolds.
2901.
SHARP ASYMPTOTICS OF THE FUNCTIONAL QUANTIZATION PROBLEM FOR GAUSSIAN
PROCESSES
Harald Luschgy and Gilles Pages
The sharp asymptotics for the L^2-quantization errors of Gaussian measures on
a Hilbert space and, in particular, for Gaussian processes is derived.
The condition imposed is regular variation of the eigenvalues.
2902.
ON WEIGHTED U-STATISTICS FOR STATIONARY PROCESSES
Tailen Hsing and Wei Biao Wu
A weighted U-statistic based on a random sample X_1,...,X_n has the form
U_n=\sum_{1\le i,j\le n}w_{i-j}K(X_i,X_j), where K is a fixed symmetric
measurable function and the w_i are symmetric weights. A large class of
statistics can be expressed as weighted U-statistics or variations thereof.
This paper establishes the asymptotic normality of U_n when the sample
observations come from a nonlinear time series and linear processes.
2903.
UNIQUENESS OF SOLUTIONS OF THE STOCHASTIC NAVIER-STOKES EQUATION WITH
INVARIANT MEASURE GIVEN BY THE ENSTROPHY
S. Albeverio and B. Ferrario
A stochastic Navier-Stokes equation with space-time Gaussian white noise is
considered, having as infinitesimal invariant measure a Gaussian measure
\mu_{\nu} whose covariance is given in terms of the enstrophy. Pathwise
uniqueness for \mu_{\nu}-a.e. initial velocity is proven for solutions having
\mu_{\nu} as invariant measure.
2904.
ON HOEFFDING'S INEQUALITIES
Vidmantas Bentkus
In a celebrated work by Hoeffding [J. Amer. Statist. Assoc. 58 (1963) 13-30],
several inequalities for tail probabilities of sums M_n=X_1+... +X_n of bounded
independent random variables X_j were proved. These inequalities had a
considerable impact on the development of probability and statistics, and
remained unimproved until 1995 when Talagrand [Inst. Hautes Etudes Sci. Publ.
Math. 81 (1995a) 73-205] inserted certain missing factors in the bounds of two
theorems. By similar factors, a third theorem was refined by Pinelis [Progress
in Probability 43 (1998) 257-314] and refined (and extended) by me. In this
article, I introduce a new type of inequality.
Namely, I show that P{M_n\geq x}\leq cP{S_n\geq x}, where c is an absolute
constant and S_n=\epsilon_1+... +\epsilon_n is a sum of independent identically
distributed Bernoulli random variables (a random variable is called Bernoulli
if it assumes at most two values). The inequality holds for those x\in R where
the survival function x\mapsto P{S_n\geq x} has a jump down. For the remaining
x the inequality still holds provided that the function between the adjacent
jump points is interpolated linearly or \log-linearly. If it is necessary, to
estimate P{S_n\geq x} special bounds can be used for binomial probabilities.
The results extend to martingales with bounded differences. It is apparent that
Theorem 1.1 of this article is the most important.
2905.
MARTINGALE APPROXIMATIONS FOR SUMS OF STATIONARY PROCESSES
Wei Biao Wu and Michael Woodroofe
Approximations to sums of stationary and ergodic sequences by martingales are
investigated. Necessary and sufficient conditions for such sums to be
asymptotically normal conditionally given the past up to time 0 are obtained.
It is first shown that a martingale approximation is necessary for such
normality and then that the sums are asymptotically normal if and only if the
approximating martingales satisfy a Lindeberg-Feller condition.
Using the explicit construction of the approximating martingales, a central
limit theorem is derived for the sample means of linear processes. The
conditions are not sufficient for the functional version of the central limit
theorem. This is shown by an example, and a slightly stronger sufficient
condition is given.
2906.
RELATIVE ENTROPY AND VARIATIONAL PROPERTIES OF GENERALIZED GIBBSIAN
MEASURES
Christof Kulske, Arnaud Le Ny and Frank Redig
We study the relative entropy density for generalized Gibbs measures. We
first show its existence and obtain a familiar expression in terms of entropy
and relative energy for a class of ``almost Gibbsian measures'' (almost sure
continuity of conditional probabilities). For quasilocal measures, we obtain a
full variational principle. For the joint measures of the random field Ising
model, we show that the weak Gibbs property holds, with an almost surely
rapidly decaying translation-invariant potential. For these measures we show
that the variational principle fails as soon as the measures lose the almost
Gibbs property. These examples suggest that the class of weakly Gibbsian
measures is too broad from the perspective of a reasonable thermodynamic
formalism.
2907.
AUXILIARY SDES FOR HOMOGENIZATION OF QUASILINEAR PDES WITH PERIODIC
COEFFICIENTS
Francois Delarue
We study the homogenization property of systems of quasi-linear PDEs of
parabolic type with periodic coefficients, highly oscillating drift and highly
oscillating nonlinear term. To this end, we propose a probabilistic approach
based on the theory of forward-backward stochastic differential equations and
introduce the new concept of ``auxiliary SDEs.''
2908.
A STOCHASTIC LOG-LAPLACE EQUATION
Jie Xiong
We study a nonlinear stochastic partial differential equation whose solution
is the conditional log-Laplace functional of a superprocess in a random
environment. We establish its existence and uniqueness by smoothing out the
nonlinear term and making use of the particle system representation developed
by Kurtz and Xiong [Stochastic Process. Appl. 83 (1999) 103-126].
We also derive the Wong-Zakai type approximation for this equation. As an
application, we give a direct proof of the moment formulas of Skoulakis and
Adler [Ann. Appl. Probab. 11 (2001) 488-543].
2909.
ASYMPTOTIC BEHAVIOR OF DIVERGENCES AND CAMERON-MARTIN THEOREM ON LOOP
SPACES
Xiang Dong Li
We first prove the L^p-convergence (p\geq 1) and a Fernique-type exponential
integrability of divergence functionals for all Cameron-Martin vector fields
with respect to the pinned Wiener measure on loop spaces over a compact
Riemannian manifold. We then prove that the Driver flow is a smooth transform
on path spaces in the sense of the Malliavin calculus and has an
\infty-quasi-continuous modification which can be quasi-surely well defined on
path spaces. This leads us to construct the Driver flow on loop spaces through
the corresponding flow on path spaces. Combining these two results with the
Cruzeiro lemma
[J. Funct. Anal. 54 (1983) 206-227] we give an alternative proof of the
quasi-invariance of the pinned Wiener measure under Driver's flow on loop
spaces which was established earlier by Driver [Trans. Amer. Math. Soc.
342 (1994) 375-394] and Enchev and Stroock [Adv. Math. 119 (1996) 127-154] by
Doob's h-processes approach together with the short time estimates of the
gradient and the Hessian of the logarithmic heat kernel on compact
Riemannian manifolds. We also establish the L^p-convergence (p\geq 1) and a
Fernique-type exponential integrability theorem for the stochastic
anti-development of pinned Brownian motions on compact Riemannian manifold with
an explicit exponential exponent. Our results generalize and sharpen some
earlier results due to Gross [J. Funct. Anal. 102 (1991) 268-313] and Hsu
[Math. Ann. 309
(1997) 331-339]. Our method does not need any heat kernel estimate and is
based on quasi-sure analysis and Sobolev estimates on path spaces.
2910.
STRONG MEMORYLESS TIMES AND RARE EVENTS IN MARKOV RENEWAL POINT
PROCESSES
Torkel Erhardsson
Let W be the number of points in (0,t] of a stationary finite-state Markov
renewal point process. We derive a bound for the total variation distance
between the distribution of W and a compound Poisson distribution. For any
nonnegative random variable \zeta, we construct a ``strong memoryless time''
\hat \zeta such that \zeta-t is exponentially distributed conditional on {\hat
\zeta\leq t, \zeta>t}, for each t. This is used to embed the Markov renewal
point process into another such process whose state space contains a frequently
observed state which represents loss of memory in the original process. We then
write W as the accumulated reward of an embedded renewal reward process, and
use a compound Poisson approximation error bound for this quantity by
Erhardsson. For a renewal process, the bound depends in a simple way on the
first two moments of the interrenewal time distribution, and on two constants
obtained from the Radon-Nikodym derivative of the interrenewal time
distribution with respect to an exponential distribution.
For a Poisson process, the bound is 0.
2911.
CHARACTERIZATION OF THE CUBIC EXPONENTIAL FAMILIES BY ORTHOGONALITY OF
POLYNOMIALS
Abdelhamid Hassairi and Mohammed Zarai
This paper introduces a notion of 2-orthogonality for a sequence of
polynomials to give extended versions of the Meixner and Feinsilver
characterization results based on orthogonal polynomials. These new versions
subsume the
Letac-Mora characterization of the real natural exponential families having
cubic variance function.
2912.
MEASURE CONCENTRATION FOR EUCLIDEAN DISTANCE IN THE CASE OF DEPENDENT
RANDOM VARIABLES
Katalin Marton
Let q^n be a continuous density function in n-dimensional Euclidean space.
We think of q^n as the density function of some random sequence X^n with
values in \BbbR^n. For I\subset[1,n], let X_I denote the collection of
coordinates X_i, i\in I, and let \bar X_I denote the collection of coordinates
X_i, i\notin I. We denote by Q_I(x_I|\bar x_I) the joint conditional density
function of X_I, given \bar X_I. We prove measure concentration for q^n in the
case when, for an appropriate class of sets I, (i) the conditional densities
Q_I(x_I|\bar x_I), as functions of x_I, uniformly satisfy a logarithmic
Sobolev inequality and (ii) these conditional densities also satisfy a
contractivity condition related to Dobrushin and Shlosman's strong mixing
condition.
2913.
STEIN'S METHOD, PALM THEORY AND POISSON PROCESS APPROXIMATION
Louis H. Y. Chen and Aihua Xia
The framework of Stein's method for Poisson process approximation is
presented from the point of view of Palm theory, which is used to construct
Stein identities and define local dependence. A general result (Theorem
\refimportantproposition) in Poisson process approximation is proved by taking
the local approach.
It is obtained without reference to any particular metric, thereby allowing
wider applicability. A Wasserstein pseudometric is introduced for measuring the
accuracy of point process approximation. The pseudometric provides a
generalization of many metrics used so far, including the total variation
distance for random variables and the Wasserstein metric for processes as in
Barbour and Brown [Stochastic Process. Appl. 43 (1992) 9-31]. Also, through the
pseudometric, approximation for certain point processes on a given carrier
space is carried out by lifting it to one on a larger space, extending an idea
of Arratia, Goldstein and Gordon [Statist. Sci. 5 (1990)
403-434]. The error bound in the general result is similar in form to that for
Poisson approximation. As it yields the Stein factor 1/\lambda as in
Poisson approximation, it provides good approximation, particularly in cases
where \lambda is large. The general result is applied to a number of problems
including Poisson process modeling of rare words in a DNA sequence.
2914.
WEIGHTED UNIFORM CONSISTENCY OF KERNEL DENSITY ESTIMATORS
Evarist Gine, Vladimir Koltchinskii and Joel Zinn
Let f_n denote a kernel density estimator of a continuous density f in d
dimensions, bounded and positive. Let \Psi(t) be a positive continuous function
such that \|\Psi f^{\beta}\|_{\infty}<\infty for some 0<\beta<1/2.
Under natural smoothness conditions, necessary and sufficient conditions for
the sequence \sqrt\frac{nh_n^d}{2|\log
h_n^d|}\|\Psi(t)(f_n(t)-Ef_n(t))\|_{\infty} to be stochastically bounded and to
converge a.s. to a constant are obtained.
Also, the case of larger values of \beta is studied where a similar sequence
with a different norming converges a.s. either to 0 or to +\infty, depending on
convergence or divergence of a certain integral involving the tail
probabilities of \Psi(X). The results apply as well to some discontinuous not
strictly positive densities.
2915.
VERTEX-REINFORCED RANDOM WALK ON Z EVENTUALLY GETS STUCK ON FIVE POINTS
Pierre Tarres
Vertex-reinforced random walk (VRRW), defined by Pemantle in 1988, is a
random process that takes values in the vertex set of a graph G, which is more
likely to visit vertices it has visited before. Pemantle and Volkov considered
the case when the underlying graph is the one-dimensional integer lattice Z.
They proved that the range is almost surely finite and that with positive
probability the range contains exactly five points. They conjectured that this
second event holds with probability 1. The proof of this conjecture is the main
purpose of this paper.
2916.
TRANSPORTATION COST-INFORMATION INEQUALITIES AND APPLICATIONS TO RANDOM
DYNAMICAL SYSTEMS AND DIFFUSIONS
H. Djellout, A. Guillin and L. Wu
We first give a characterization of the L^1-transportation cost-information
inequality on a metric space and next find some appropriate sufficient
condition to transportation cost-information inequalities for dependent
sequences. Applications to random dynamical systems and diffusions are studied.
2917.
TWO-PLAYER NONZERO-SUM STOPPING GAMES IN DISCRETE TIME
Eran Shmaya and Eilon Solan
We prove that every two-player nonzero-sum stopping game in discrete time
admits an \epsilon-equilibrium in randomized strategies for every \epsilon >0.
We use a stochastic variation of Ramsey's theorem, which enables us to reduce
the problem to that of studying properties of \epsilon-equilibria in a simple
class of stochastic games with finite state space.
2918.
LARGE DEVIATION ASYMPTOTICS FOR OCCUPANCY PROBLEMS
Paul Dupuis, Carl Nuzman and Phil Whiting
In the standard formulation of the occupancy problem one considers the
distribution of r balls in n cells, with each ball assigned independently to a
given cell with probability 1/n. Although closed form expressions can be given
for the distribution of various interesting quantities (such as the fraction of
cells that contain a given number of balls), these expressions are often of
limited practical use. Approximations provide an attractive alternative, and in
the present paper we consider a large deviation approximation as r and n tend
to infinity. In order to analyze the problem we first consider a dynamical
model, where the balls are placed in the cells sequentially and ``time''
corresponds to the number of balls that have already been thrown. A complete
large deviation analysis of this ``process level'' problem is carried out, and
the rate function for the original problem is then obtained via the contraction
principle. The variational problem that characterizes this rate function is
analyzed, and a fairly complete and explicit solution is obtained. The
minimizing trajectories and minimal cost are identified up to two constants,
and the constants are characterized as the unique solution to an elementary
fixed point problem. These results are then used to solve a number of
interesting problems, including an overflow problem and the partial coupon
collector's problem.
2919.
LARGE DEVIATIONS FOR RANDOM POWER MOMENT PROBLEM
Fabrice Gamboa and Li-Vang Lozada-Chang
We consider the set M_n of all n-truncated power moment sequences of
probability measures on [0,1]. We endow this set with the uniform probability.
Picking randomly a point in M_n, we show that the upper canonical measure
associated with this point satisfies a large deviation principle. Moderate
deviation are also studied completing earlier results on asymptotic normality
given by \citeauthorChKS93 [Ann. Probab. 21 (1993) 1295-1309]. Surprisingly,
our large deviations results allow us to compute explicitly the (n+1)th moment
range size of the set of all probability measures having the same n first
moments. The main tool to obtain these results is the representation of M_n on
canonical moments [see the book of \citeauthorDS97].
2920.
ON THE CONTRACTION METHOD WITH DEGENERATE LIMIT EQUATION
Ralph Neininger and Ludger Ruschendorf
A class of random recursive sequences (Y_n) with slowly varying variances as
arising for parameters of random trees or recursive algorithms leads after
normalizations to degenerate limit equations of the form X\stackrel{L}{=}X.
For nondegenerate limit equations the contraction method is a main tool to
establish convergence of the scaled sequence to the ``unique'' solution of the
limit equation. In this paper we develop an extension of the contraction method
which allows us to derive limit theorems for parameters of algorithms and data
structures with degenerate limit equation. In particular, we establish some new
tools and a general convergence scheme, which transfers information on mean and
variance into a central limit law (with normal limit). We also obtain a
convergence rate result. For the proof we use selfdecomposability properties of
the limit normal distribution which allow us to mimic the recursive sequence by
an accompanying sequence in normal variables.
2921.
ON DIFFUSIVITY OF A TAGGED PARTICLE IN ASYMMETRIC ZERO-RANGE DYNAMICS
Sunder Sethuraman
Consider a tagged particle in zero-range dynamics on the integer lattice in
dimension d with rate g whose finite-range jump probabilities p possess a
drift. We show, in equilibrium, that the variance of the tagged particle
position at time t is at least order t in all dimensions and at most order t in
d=1 and d larger or equal to 3 for a wide class of rates g. Also, in d=1, when
the jump distribution p is totally asymmetric and nearest-neighbor, and when
the rate g(k) increases and g(k)/k decreases with k, we show the diffusively
scaled centered tagged particle position converges to a Brownian motion.
2922.
SUPERDIFFUSIVITY OF OCCUPATION-TIME VARIANCE IN 2-DIMENSIONAL ASYMMETRIC
PROCESSES WITH DENSITY 1/2
Sunder Sethuraman
We compute that the growth of the origin occupation-time variance up to time
t in dimension d=2 with respect to asymmetric simple exclusion in equilibrium
with density 1/2 is in a certain sense at least t(log(log t)) for general
rates, and at least t(log t)^{1/2} for rates which are asymmetric only in the
direction of one of the axes. These estimates are consistent with conjectures
with respect to the transition function and variance of 'second-class'
particles.
2923.
SPATIAL BIRTH-AND-DEATH PROCESSES IN RANDOM ENVIRONMENT
Roberto Fernandez, Pablo A. Ferrari, Gustavo R. Guerberoff
We consider birth-and-death processes of objects (animals) defined in ${\bf
Z}^d$ having unit death rates and random birth rates. For animals with
uniformly bounded diameter we establish conditions on the rate distribution
under which the following holds for almost all realizations of the birth rates:
(i) the process is ergodic with at worst power-law time mixing; (ii) the unique
invariant measure has exponential decay of (spatial) correlations; (iii) there
exists a perfect-simulation algorithm for the invariant measure. The results
are obtained by first dominating the process by a backwards oriented
percolation model, and then using a multiscale analysis due to Klein to
establish conditions for the absence of percolation.
2924.
MARTINGALES AND PROFILE OF BINARY SEARCH TREES
Brigitte Chauvin, Thierry Klein, Jean-Francois Marckert, Alain Rouault
We are interested in the asymptotic analysis of the binary search tree (BST)
under the random permutation model. Via an embedding in a continuous time
model, we get new results, in particular the asymptotic behavior of the
profile.
2925.
TIMESCALES OF POPULATION RARITY AND COMMONNESS IN RANDOM ENVIRONMENTS
R. Ferriere, A. Guionnet, I. Kurkova
This paper investigates the influence of environmental noise on the
characteristic timescale of the dynamics of density-dependent populations.
General results are obtained on the statistics of time spent in rarity and time
spent in commonness. The nonlinear stochastic models under consideration form a
class of Markov chains on the state space $]0, \infty[$ which are transient if
the intrinsic growth rate is negative and recurrent if it is positive or null.
In the recurrent case, we obtain a necessary and sufficient condition for
positive recurrence and precise estimates for the distribution of times of
rarity and commonness. In the null recurrent, critical case that applies to
ecologically neutral species, the distribution of rarity time is a universal
power law with exponent -3/2. These non- trivial results should be of interest
to biologists involved in the conservation of threatened populations, and to
epidemiologists facing the need to better understanding the dynamics of pest or
disease outbreaks.
2926.
CAPACITIES IN WIENER SPACE, QUASI-SURE LOWER FUNCTIONS, AND KOLMOGOROV'S
EPSILON-ENTROPY
Davar Khoshnevisan, David A. Levin, Pedro J. Mendez-Hernandez
We propose a set-indexed family of capacities $\{\cap_G \}_{G \subseteq
\R_+}$ on the classical Wiener space $C(\R_+)$. This family interpolates
between the Wiener measure ($\cap_{\{0\}}$) on $C(\R_+)$ and the standard
capacity ($\cap_{\R_+}$) on Wiener space. We then apply our capacities to
characterize all quasi-sure lower functions in $C(\R_+)$. In order to do this
we derive the following capacity estimate which may be of independent interest:
There exists a constant $a > 1$ such that for all $r > 0$,
\[
\frac {1}{a} \K_G(r^6) e^{-\pi^2/(8r^2)} \le \cap_G \{f^* \le r\}
\le a \K_G(r^6) e^{-\pi^2/(8r^2)}.
\]
Here, $\K_G$ denotes the Kolmogorov $\epsilon$-entropy of $G$, and $f^* :=
\sup_{[0,1]}|f|$.
2927.
CHARACTERIZATION OF INVARIANT MEASURES AT THE LEADING EDGE FOR COMPETING
PARTICLE SYSTEMS
A. Ruzmaikina, M. Aizenman
We study systems of particles on a line which have a maximum, are locally
finite, and evolve with independent increments. 'Quasi-stationary states' are
defined as probability measures, on the sigma-algebra generated by the gap
variables, for which the joint distribution of the gaps is invariant under the
time evolution. Examples are provided by Poisson processes with exponential
densities and linear superpositions of such measures. We show that conversely:
any quasi-stationary state for the independent dynamics, with an exponentially
bounded integrated density of particles, corresponds to a superposition of the
above described probability measures, restricted to the relevant sigma-algebra.
Among the system for which this question is of some relevance are spin-glass
models of statistical mechanics, where the point process represents the
collection of the free energies of the distinct pure states, the time evolution
corresponds to the addition of the spin variable, and the Poisson measures
described above correspond to the so-called REM states.
2928.
BALANCED BOOLEAN FUNCTIONS THAT CAN BE EVALUATED SO THAT EVERY INPUT BIT
IS UNLIKELY TO BE READ
Itai Benjamini, Oded Schramm, David B. Wilson
A Boolean function of n bits is balanced if it takes the value 1 with
probability 1/2. We exhibit a balanced Boolean function with a randomized
evaluation procedure (with probability 0 of making a mistake) so that on
uniformly random inputs, no input bit is read with probability more than
Theta(n^{-1/2} sqrt{log n}). We give a balanced monotone Boolean function for
which the corresponding probability is Theta(n^{-1/3} log n). We then show that
for any randomized algorithm for evaluating a balanced Boolean function, when
the input bits are uniformly random, there is some input bit that is read with
probability at least Theta(n^{-1/2}). For balanced monotone Boolean functions,
there is some input bit that is read with probability at least Theta(n^{-1/3}).
2929.
ON THE SMOOTH-FIT PROPERTY FOR ONE-DIMENSIONAL OPTIMAL SWITCHING PROBLEM
Huyen Pham
This paper studies the problem of optimal switching for one-dimensional
diffusion, which may be regarded as sequential optimal stopping problem with
changes of regimes. The resulting dynamic programming principle leads to a
system of variational inequa-lities, and the state space is divided into
continuation regions and switching regions. By means of viscosity solutions
approach, we prove the smoot-fit $C^1$ property of the value functions.
2930.
CONTINUOUS TIME MARKOV PROCESSES ON GRAPHS
Jianjun Tian and Xiao-Song Lin
We study continuous time Markov processes on graphs. The notion of frequency
is introduced, which serves well as a scaling factor between any Markov time of
a continuous time Markov process and that of its jump chain. As an application,
we study ``multi-person simple random walks'' on a graph G with n vertices.
There are n persons distributed randomly at the vertices of G. In each step of
this discrete time Markov process, we randomly pick up a person and move it to
a random adjacent vertex. We give estimate on the expected number of steps for
these $n$ persons to meet all together at a specific vertex, given that they
are at different vertices at the begininng. For regular graphs, our estimate is
exact.
2931.
BRANCHING PROCESSES, AND RANDOM-CLUSTER MEASURES ON TREES
Geoffrey Grimmett, Svante Janson
Random-cluster measures on infinite regular trees are studied in conjunction
with a general type of `boundary condition', namely an equivalence relation on
the set of infinite paths of the tree. The uniqueness and non-uniqueness of
random-cluster measures are explored for certain classes of equivalence
relations. In proving uniqueness, the following problem concerning branching
processes is encountered and answered. Consider bond percolation on the
family-tree $T$ of a branching process. What is the probability that every
infinite path of $T$, beginning at its root, contains some vertex which is
itself the root of an infinite open sub-tree?
2932.
CONNECTING YULE PROCESS, BISECTION AND BINARY SEARCH TREE VIA
MARTINGALES
B. Chauvin and A. Rouault
We present new links between some remarkable martingales found in the study
of the Binary Search Tree, or of the Bisection Problem, looking at them on the
probability space of a continuous time binary branching process.
2933.
MARKOV CHAIN COMPARISON
Martin Dyer, Leslie Ann Goldberg, Mark Jerrum, Russell Martin
This is an expository paper, focussing on the following scenario. We have two
Markov chains, M and M'. By some means, we have obtained a bound on the mixing
time of M'. We wish to compare M with M' in order to derive a corresponding
bound on the mixing time of M. We investigate the application of the comparison
method of Diaconis and Saloff-Coste to this scenario, giving a number of
theorems which characterize the applicability of the method. We focus
particularly on the case in which the chains are not reversible. The purpose of
the paper is to provide a catalogue of theorems which can be easily applied to
bound mixing times.
2934.
THE CRITICAL PROBABILITY FOR RANDOM VORONOI PERCOLATION IN THE PLANE IS
1/2
Bela Bollobas and Oliver Riordan
We study percolation in the following random environment: let $Z$ be a
Poisson process of constant intensity in the plane, and form the Voronoi
tessellation of the plane with respect to $Z$. Colour each Voronoi cell black
with probability $p$, independently of the other cells. We show that the
critical probability is 1/2. More precisely, if $p>1/2$ then the union of the
black cells contains an infinite component with probability 1, while if $p<1/2$
then the distribution of the size of the component of black cells containing a
given point decays exponentially. These results are analogous to Kesten's
results for bond percolation in the square lattice.
The result corresponding to Harris' Theorem for bond percolation in the
square lattice is known: Zvavitch noted that one of the many proofs of this
result can easily be adapted to the random Voronoi setting. For Kesten's
results, none of the existing proofs seems to adapt. The methods used here also
give a new and very simple proof of Kesten's Theorem for the square lattice; we
hope they will be applicable in other contexts as well.
2935.
A SHORT PROOF OF THE HARRIS-KESTEN THEOREM
Bela Bollobas and Oliver Riordan
We give a short proof of the fundamental result that the critical probability
for bond percolation in the planar square lattice is equal to 1/2. The upper
bound was proved by Harris, who showed in 1960 that percolation does not occur
at $p=1/2$. The other, more difficult, bound was proved by Kesten, who showed
in 1980 that percolation does occur for any $p>1/2$.
2936.
A PHASE TRANSITION IN A MODEL FOR THE SPREAD OF AN INFECTION
Harry Kesten and Vladas Sidoravicius
We show that a certain model for the spread of an infection has a phase
transition in the recuperation rate. The model is as follows: There are
particles or individuals of type A and type B, interpreted as healthy and
infected, respectively. All particles perform independent, continuous time,
simple random walks on Z^d with the same jump rate D. The only interaction
between the particles is that at the moment when a B-particle jumps to a site
which contains an A-particle, or vice versa, the A-particle turns into a
B-particle. All B-particles recuperate (that is, turn back into A-particles)
independently of each other at a rate lamda. We assume that we start the system
with N_A(x,0-) A-particles at x, and that the N_A(x,0-), x in Z^d, are i.i.d.,
mean mu_A Poisson random variables. In addition we start with one additional
B-particle at the origin. We show that there is a critical recuperation rate
lambda_c > 0 such that the B-particles survive (globally) with positive
probability if lambda < lamda_c and die out with probability 1 if lambda >
\lamda_c.
2937.
A CRITICAL BRANCHING PROCESS MODEL FOR BIODIVERSITY
David J. Aldous and Lea Popovic
Motivated as a null model for comparison with data, we study the following
model for a phylogenetic tree on $n$ extant species. The origin of the clade is
a random time in the past, whose (improper) distribution is uniform on
$(0,\infty)$. After that origin, the process of extinctions and speciations is
a continuous-time critical branching process of constant rate, conditioned on
having the prescribed number $n$ of species at the present time. We study
various mathematical properties of this model as $n \to \infty$ limits: time of
origin and of most recent common ancestor; pattern of divergence times within
lineage trees; time series of numbers of species; number of extinct species in
total, or ancestral to extant species; and "local" structure of the tree
itself. We emphasize several mathematical techniques: associating walks with
trees, a point process representation of lineage trees, and Brownian limits.
2938.
HITTING PROPERTIES OF S.P.D.E.'S WITH REFLECTION
Robert C. Dalang, Carl Mueller, Lorenzo Zambotti
We study the hitting properties of the solutions $u$ of a class of stochastic
p.d.e.'s with singular drifts that prevent $u$ from becoming negative. The
drifts can be a reflecting term or a non-linearity $c u^{-3}$, with $c>0$. We
prove that almost surely, for all time $t>0$, the solution $u_t$ hits the level
0 only at a finite number of space points, which depends explicitly on $c$. In
particular, this number of hits never exceeds 4, and if $c> 15/8$, then level 0
is not hit.
2939.
SCALING LIMITS OF THE UNIFORM SPANNING TREE AND LOOP-ERASED RANDOM WALK
ON FINITE GRAPHS
Yuval Peres and David Revelle
Let x and y be chosen uniformly in a graph G. We find the limiting
distribution of the length of a loop-erased random walk from x to y on a large
class of graphs that include the discrete torus in dimensions 5 and above.
Moreover, on this family of graphs we show that a suitably normalized
finite-dimensional scaling limit of the uniform spanning tree is a Brownian
continuum random tree.
2940.
ON THE SYMMETRY OF THE DIFFUSION COEFFICIENT IN ASYMMETRIC SIMPLE
EXCLUSION
Michail Loulakis
We prove the symmetry of the Diffusion Coefficient appearing in the
fluctuation-dissipation theorem for the general asymmetric simple exclusion
process.
2941.
DYNAMIC MONETARY RISK MEASURES FOR BOUNDED DISCRETE-TIME PROCESSES
Patrick Cheridito, Freddy Delbaen, Michael Kupper
We study time-consistency questions for processes of monetary risk measures
that depend on bounded discrete-time processes describing the evolution of
financial values. The time horizon can be finite or infinite. We call a process
of monetary risk measures time-consistent if it assigns to a process of
financial values the same risk irrespective of whether it is calculated
directly or in two steps backwards in time, and we show how this property
manifests itself in the corresponding process of acceptance sets. For processes
of coherent and convex monetary risk measures admitting a robust representation
with sigma-additive linear functionals, we give necessary and sufficient
conditions for time-consistency in terms of the representing functionals.
2942.
LARGE DEVIATIONS FOR WISHART PROCESSES
Catherine Donati-Martin
Let $X^{(\delta)}$ be a Wishart process of dimension $\delta$, with values in
the set of positive matrices of size $m$. We are interested in the large
deviations for a family of matrix-valued processes $\{\delta^{-1}
X_t^{(\delta)}, t \leq 1 \}$ as $\delta$ tends to infinity. The process
$X^{(\delta)}$ is a solution of a stochastic differential equation with a
degenerate diffusion coefficient. Our approach is based upon the introduction
of exponential martingales. We give some applications to large deviations for
functionals of the Wishart processes, for example the set of eigenvalues.
2943.
TAYLOR EXPANSIONS OF R-TRANSFORMS, APPLICATION TO SUPPORTS AND MOMENTS
Florent Benaych-Georges
We prove that a probability measure on the real line has a moment of order p
(even integer), if and only if its R-transform admits a Taylor expansion with p
terms. We also prove a weaker version of this result when p is odd. We then
apply this to prove that a probability measure whose R-transform extends
analytically to a ball with center zero is compactly supported, and that a free
infinitely divisible distribution has a moment of order p even, if and only if
its Levy measure does so. We also prove a weaker version of the last result
when p is odd.
2944.
SCALING LIMIT AND CRITICAL EXPONENTS FOR TWO-DIMENSIONAL BOOTSTRAP
PERCOLATION
Federico Camia
Consider a cellular automaton with state space $\{0,1 \}^{{\mathbb Z}^2}$
where the initial configuration $\omega_0$ is chosen according to a Bernoulli
product measure, 1's are stable, and 0's become 1's if they are surrounded by
at least three neighboring 1's. In this paper we show that the configuration
$\omega_n$ at time n converges exponentially fast to a final configuration
$\bar\omega$, and that the limiting measure corresponding to $\bar\omega$ is in
the universality class of Bernoulli (independent) percolation.
More precisely, assuming the existence of the critical exponents $\beta$,
$\eta$, $\nu$ and $\gamma$, and of the continuum scaling limit of crossing
probabilities for independent site percolation on the close-packed version of
${\mathbb Z}^2$ (i.e., for independent $*$-percolation on ${\mathbb Z}^2$), we
prove that the bootstrapped percolation model has the same scaling limit and
critical exponents.
This type of bootstrap percolation can be seen as a paradigm for a class of
cellular automata whose evolution is given, at each time step, by a monotonic
and nonessential enhancement.
2945.
RELATIVISTIC DIFFUSIONS AND SCHWARZSCHILD SPACE
Jacques Franchi and Yves Le Jan
The purpose of this article is to introduce and study a relativistic motion
whose acceleration, in proper time, is given by a white noise. Beginning with
the flat case of special relativity, we deal with the case of general
relativity, and we finally consider closely the Schwarzschild geometry example.
2946.
PROPERTIES OF CONVOLUTIONS ARISING IN STOCHASTIC VOLTERRA EQUATIONS
Anna Karczewska
The aim of the paper is to provide some regularity results for stochastic
convolutions corresponding to stochastic Volterra equations in separable
Hilbert space. We study convolutions of the form $W^{\Psi}(t):=\int_0^t
S(t-\tau)\Psi(\tau)dW(\tau)$, $t\geq 0$, where $S(t), t\geq 0$, is so-called
{\em resolvent} for Volterra equation considered,$\Psi$ is an appropriate
process and $W$ is a cylindrical Wiener process. In the paper we extend the
semigroup approach to stochastic convolutions with resolvent operators.
2947.
COLORED GENEALOGICAL TREES AND COALESCENT THEORY
Jianjun Tian and Xiao-Song Lin
We introduce a colored coalescent process which recovers random colored
genealogical trees. Here a colored genealogical tree has its vertices colored
black or white. Moving backward along the colored genealogical tree, the color
of vertices may change only when two vertice coalesce. The rule that governs
the change of color involves a parameter $x$. When $x=1/2$, the colored
coalescent process can be derived from a variant of the Wright-Fisher model for
a haploid population in population genetics. Explicit computations of the
expectation and the cumulative distribution function of the coalescent time are
carried out. For example, our calculation shows that when $x=1/2$, for a sample
of $n$ colored individuals, the expected time for the colored coalescent
process to reach a black MRAC or a white MRAC, respectively, is $3-2/n$. On the
other hand, the expected time for the colored coalescent process to reach a
MRAC, either black or white, is $2-2/n$, which is the same as that for the
standard Kingman coalescent process. This colored coalescent process with a
color mutation process superimposed is also studied in explicit details.
2948.
RANDOM TREE GROWTH WITH GENERAL WEIGHT FUNCTION
Anna Rudas
We extend the results of B. Bollobas, O. Riordan, J. Spencer, G. Tusnady, and
Mori. We consider a model of random tree growth, where at each time unit a new
node is added and attached to an already existing node chosen at random. The
probability with which a node with degree $k$ is chosen is proportional to
$w(k)$, where $w$ is a fixed weight function. We prove that if $w$ fulfills
some asymptotic requirements then the degree sequence converges in probability,
we give the limit. In particular if $w$ is asymptotically linear then the
degree sequence decays with power law. Our method of proof is analytic rather
than combinatorial, having the advantage of robustness: only asymptotic
properties of the weight function $w$ are used, while in the cited papers the
explicit law $w(k)=ak+b$ is assumed.
2949.
ON MEASURES OF UNFAIRNESS AND AN OPTIMAL CURRENCY TRANSACTION TAX
Frederik Herzberg
Firstly, we will describe a model of how herd behaviour and self-fulfilling
prophecies can influence currency exchange rates, and what the impact of a
currency transaction tax would be. These considerations yield a stochastic
differential equation, whose solution will never be a martingale unless the tax
level prevents any transactions whatsoever. We will show, using a suitable
notion of unfairness for discounted price processes that the fairest tax rate
is the maximal one subject to the condition that it does not affect
real-economic speculation.
2950.
THE FAIREST PRICE OF AN ASSET IN AN ENVIRONMENT OF TEMPORARY ARB
