We consider a stationary sequence of associated real random variables and state conditions which guarantee that partial sums of this sequence, when properly normalized, converge in distribution to a stable, non-Gaussian limit. Limit theorems for jointly stable and associated random variables are investigated in detail. In the general case we assume that finite-dimensional distributions belong to the domain of attraction of multidimensional strictly stable laws and that there is a bound on the positive dependence given by finiteness of an analog to the lag covariance series.

Let $(X_n)_{n\geq 1}$ be a sequence of i.i.d. r.v.'s with values in a measurable space $(E, \mathscr{E})$ of law $\mu$, and consider the empirical process $L_n(f) = (1/n)\sum^n_{k=1} f(X_k)$ with $f$ varying in a class of bounded functions $\mathscr{F}$. Using a recent isoperimetric inequality of Talagrand, we obtain the necessary and sufficient conditions for the large deviation estimations, the moderate deviation estimations and the LIL of $L_n(\cdot)$ in the Banach space of bounded functionals $\mathscr{l}_\infty(\mathscr{F})$. The extension to the unbounded functionals is also discussed.

Under natural conditions on a class $\mathscr{F}$ of functions on a probability space, near optimal bounds are given for the probabilities $P\big(\sup_{f\in\mathscr{F}}|\sum_{i\leq n} f(X_i) - nE(f)| \geq M\sqrt n\big)$. The method is a variation of this author's method to study the tail probability of the supremum of a Gaussian process.

We give a complete and rather explicit characterization of the upper and lower classes for a family of stationary Gaussian stochastic processes.

Classical empirical process theory for Vapnik-Cervonenkis classes deals mainly with sequences of independent variables. This paper extends the theory to stationary sequences of dependent variables. It establishes rates of convergence for $\beta$-mixing and $\phi$-mixing empirical processes indexed by classes of functions. The method of proof depends on a coupling of the dependent sequence with sequences of independent blocks, to which the classical theory can be applied. A uniform $O(n^{-s/(1+s)})$ rate of convergence over V-C classes is established for sequences whose mixing coefficients decay slightly faster than $O(n^{-s})$.

It is proved that, for $h$ measurable and symmetric in its arguments and $X_i$ i.i.d., if the sequence $\{n^{-m/2} \sum_{i_1,\ldots,i_m\leq n,i_j\neq i_k \text{if} j\neq k} h(X_{i_1},\ldots, X_{i_m})\}^\infty_{n=1}$ is stochastically bounded, then $Eh^2 < \infty$ and $Eh(X_1,x_2,\ldots,x_m) = 0$ a.s.

For a sequence of bivariate pairs $(X_i, Y_i)$, the concomitant $Y_{\lbrack i\rbrack}$ of the $i$th largest $x$-value $X_{(i)}$ equals that value of $Y$ paired with $X_{(i)}$. In assessing the quality of a file-merging or file-matching procedure, the penalty for incorrect matching may often be expressed as the average value of a function of the difference $Y_{\lbrack i\rbrack} - Y_{(i)}$. We establish strong laws and central limit theorems for such quantities. Our proof is based on the observation that if $G_x(\cdot)$ denotes the distribution function of $Y$ given $X = x$, then $G_X(Y)$ is stochastically independent of $X$, even though $G_x(\cdot)$ depends numerically on $x$.

Let $X_{1,n} \leq \cdots \leq X_{n,n}$ be the order statistics of $n$ independent random variables with a common distribution function $F$ and let $k_n$ be positive numbers such that $k_n \rightarrow \infty$ and $k_n/n \rightarrow 0$ as $n \rightarrow \infty$, and consider the sums $I_n(a, b) = \sum^{\lbrack bk_n\rbrack}_{i=\lbrack ak_n\rbrack+1} X_{n+1-i,n}$ of intermediate order statistics, where $0 < a < b$. We find necessary and sufficient conditions for the existence of constants $A_n > 0$ and $C_n$ such that $A^{-1}_n(I_n(a,b) - C_n)$ converges in distribution along subsequences of the positive integers $\{n\}$ to nondegenerate limits and completely describe the possible subsequential limiting distributions.

Let $W(t)$ be the Wiener sausage in $\mathbb{R}^d$, that is, the $a$-neighborhood for some $a > 0$ of the path of Brownian motion up to time $t$. It is shown that integrals of the type $\int^t_0\nu(s) d|W(s)|$, with $t \rightarrow \nu (t)$ nonincreasing and $nu (t) \sim \nu t^{-\gamma}, t \rightarrow \infty$, have a large deviation behavior similar to that of $|W(t)|$ established by Donsker and Varadhan. Such a result gives information about the survival asymptotics for Brownian motion in a Poisson field of spherical traps of radius $a$ when the traps decay independently with lifetime distribution $\nu(t)/\nu(0)$. There are two critical phenomena: (i) in $d \geq 3$ the exponent of the tail of the survival probability has a crossover at $\gamma = 2/d$; (ii) in $d \geq 1$ the survival strategy changes at time $s = \lbrack\gamma/(1 + \gamma)\rbrack t$, provided $\gamma < 1/2, d = 1$, respectively, $\gamma < 2/d, d \geq 2$.

By using an appropriate martingale, we compute the joint distribution of the hitting time and place of a sphere by $d$-dimensional Brownian motion, when the process starts outside the sphere.

We define a notion of stochastic domination between trees, where one tree dominates another if, when the vertices of each are labeled with independent, identically distributed random variables, one tree is always more likely to contain a path with a specified property. Sufficient conditions for this kind of domination are (1) more symmetry and (2) earlier branching. We apply these conditions to the problem of determining how fast a tree must grow before first-passage percolation on the tree exhibits an explosion, that is to say, infinitely many vertices are reached in finite time. For a tree in which each vertex at distance $n - 1$ from the root has $f(n)$ offspring, $f$ nondecreasing, an explosion occurs with exponentially distributed passage times if and only if $\sum f(n)^{-1} < \infty$.

Let $T$ be the set of vertices of a homogeneous tree and let $(X_t)_{t\in T}$ be a second-order real or complex-valued process such that the expected value $\mathbb{E}(X_s\bar{X}_t)$ depends only on the distance between the vertices $s$ and $t$. In this paper we construct a measure space $(K, \mathscr{H}, m)$ and an isometry of the closed subspace of $L^2_\mathbb{C}(\Omega, \mathscr{A}, P)$ spanned by $(X_t)_{t\in T}$ onto $L^2(m)$.

We study a variant of branching Markov chains in which the branching is governed by a fixed deterministic tree $T$ rather than a Galton-Watson process. Sample path properties of these chains are determined by an interplay of the tree structure and the transition probabilities. For instance, there exists an infinite path in $T$ with a bounded trajectory iff the Hausdorff dimension of $T$ is greater than $\log(1/\rho)$ where $\rho$ is the spectral radius of the transition matrix.

Cellular automata have been the subject of considerable recent study in the statistical physics literature, where they provide examples of easily accessible nonlinear phenomena. We investigate a class of nearest neighbor cellular automata taking values $\{0,1\}$ on $\mathbb{Z}$. In the deterministic setting, this class includes rules which yield fractal-like patterns when starting from a single occupied site. We are interested here in the asymptotic behavior of systems subjected to small random perturbations. In this context, one wishes to ascertain under which conditions such systems survive with positive probability. We show here that, except in trivial cases, these systems in fact always survive, and they possess densities which remain bounded away from 0.

We consider a special class of attractive critical processes based on the transition function of a transient random walk on $\mathbb{Z}^d$. These processes have infinitely many invariant distributions and no spectral gap. The exponential $L^2$ decay is replaced by an algebraic $L^2$ decay. The paper shows the dependence of this algebraic rate in terms of the dimension of the lattice and the locality of the functions under consideration. The theory is illustrated by several examples dealing with locally interacting diffusion processes and independent random walks.

The first time that the $N$ sites to the right of the origin become empty in a one-dimensional zero-range process is shown to converge exponentially fast, as $N \rightarrow \infty$, to the exponential distribution, when divided by its mean. The initial distribution of the process is assumed to be one of the extremal invariant measures $\nu_\rho, \rho \in (0, 1)$, with density $\rho/(1 - \rho)$. The proof is based on the classical Burke theorem.

In this paper we will consider translation invariant finite range particle systems with state space $\{0, 1,\ldots,\kappa - 1\}^S$ with $S = \varepsilon \mathbb{Z}^d$. De Masi, Ferrari and Lebowitz have shown that if we introduce stirring at rate $\varepsilon^{-2}$, then the system converges to the solution of an associated reaction diffusion equation. We exploit this connection to prove results about the existence of phase transitions when the stirring rate is large that apply to a wide variety of examples with state space $\{0, 1\}^S$.

We consider a one-dimensional Glauber-Kawasaki process which gives rise in the hydrodynamical limit to a reaction diffusion equation with a double-well potential. We study the case when the process starts off from a product measure with zero averages, which, hydrodynamically, corresponds to a stationary unstable state. We prove that at times longer than the hydrodynamical ones the reaction diffusion equation no longer describes the behavior of the system, which in fact leaves the unstable equilibrium. The spatial patterns of the typical configurations when this happens are investigated.

General conditions on general state space Markov jump processes are established for the asymptotic exponentiality of the distribution of the exit time into any small forbidden set. The error bounds for this exponential approximation tend to 0 as the size of the forbidden set tends to 0.

We analyze a random walk on the orthogonal group SO$(N)$ given by repeatedly rotating by a fixed angle through randomly chosen planes of $\mathbb{R}^N$. We derive estimates of the rate at which this random walk will converge to Haar measure on SO$(N)$, using character theory and the upper bound lemma of Diaconis and Shashahani. In some cases we are able to establish the existence of a "cut off phenomenon" for the random walk. This is the first such non-trivial result on a nonfinite group.

Let $X_1$ be a $(d \times d)$ random stochastic matrix such that the rows of $X_1$ are independent, with Dirichlet distributions. The rows of the $(d \times d)$ matrix $A$ are the parameters of these Dirichlet distributions, and we assume that the sums of the rows and columns of $A$ provide the same vector $r = (r_1,\ldots,r_d)$. If $(X_n)^\infty_{n=1}$ are i.i.d., we prove that $\lim_{n\rightarrow\infty}(X_n \cdots X_1)$ almost surely has identical rows, which are Dirichlet distributed with parameter $r$. Van Assche has proved this for $d = 2$ and four identical entries for $A$.

This note studies the deterministic flow of measures which is the limiting case as $n \rightarrow \infty$ of Dyson's model of the motion of the eigenvalues of random symmetric $n \times n$ matrices. Though this flow is nonlinear, highly singular and apparently of Wiener-Hopf type, it may be solved explicitly without recourse to Wiener-Hopf theory. The solution greatly clarifies the role of the famous Wigner semicircle law.

A proof is given of a conjecture, due to F. W. Steutel: the set of positive $\alpha$-values, for which any positive mixture of $\Gamma(\alpha)$-distributions is infinitely divisible, is the interval (0, 2].

It is known that if $(X_n)$ and $(Y_n)$ are two $(\mathscr{F}_n)$-adapted sequences of random variables such that for each $k \geq 1$ the conditional distributions of $X_k$ and $Y_k$, given $\mathscr{F}_{k-1}$, coincide a.s., then the following is true: $\|\sum X_k\|_p \leq B_p\| \sum Y_k\|_p, 1 \leq p < \infty$, for some constant $B_p$ depending only on $p$. The aim of this paper is to show that if a sequence $(Y_n)$ is conditionally independent, then the constant $B_p$ may actually be chosen to be independent of $p$. This significantly improves all hitherto known estimates on $B_p$ and extends an earlier result of Klass on randomly stopped sums of independent random variables as well as our recent result dealing with martingale transforms of Rademacher sequences.

We present a general criterion for the existence of an occupation density, which is based on the integration by parts formula on Wiener space. This criterion is applied to two particular examples of anticipating processes. First we discuss the case of Brownian motion plus a nonadapted absolutely continuous drift, and second we consider the case of a Skorohod integral process. Finally a version of Tanaka's formula is proved for the Skorohod integral process.

We extend the forward-backward martingale approach to Stratonovich integrals developed by Zheng and Lyons to the general context of Dirichlet spaces. From this perspective, it is clear that the Stratonovich integral of an $L^2$ 1-form against a Dirichlet process is well defined, coordinate invariant, and obeys appropriate chain rules. The paper continues by examining the tightness and continuity of the mapping from Dirichlet forms to probability measures on path space. Some positive results are obtained for a class of infinite-dimensional diffusions.