We examine two standard types of connectivity functions in the high-density phase of nearest-neighbor Bernoulli (bond) percolation. We show that these two quantities decay exponentially at the same constant rate. The reciprocal of this constant defines therefore a correlation length. Unfortunately, we cannot prove that this correlation length is finite whenever $p > p_c$, although previous work established this result for $p$ above a threshold which is conjectured to coincide with $p_c$. We examine also a third connectivity function and prove that it too decays exponentially with the same rate as the two standard connectivity functions. We establish various useful properties of our correlation length, such a semicontinuity as a function of bond density and convexity in its directional dependence. Finally, for bond percolation in two dimensions we show that the correlation length at bond density $p_1 > p_c = \frac{1}{2}$ is exactly half the correlation length at the subcritical bond density $p_2 = 1 - p_1 < p_c$. This sharpens some other exact results for two-dimensional percolation and is the precise analog of known results for the two-dimensional Ising model.

We show that if $\sigma_N$ is the time that the contact process on $\{1, \ldots N\}$ first hits the empty set then for $\lambda = \lambda_c$, the critical value for the contact process on $\mathbb{Z}, \sigma_N/N \rightarrow \infty$ and $\sigma_N/N^4 \rightarrow 0$ in probability as $N \rightarrow \infty$. The keys to the proof are a new renormalized bond construction and lower bounds for the fluctuations of the right edge. As a consequence of the result we get bounds on some critical exponents. We also study the analogous problem for bond percolation in $\{1,\ldots N\} \times \mathbb{Z}$ and investigate the limit distribution of $\sigma_N/E\sigma_N$.

This article contains a proof of the conjectured central limit theorem for the growth of the right-hand edge for supercritical oriented percolation. The technique of proof, finding points with regeneration-type properties termed "break points," may also apply to other processes.

Let $\eta_t$ be the basic voter model on $\mathbb{Z}^d$ and let $\eta^{(N)}_t$ be the voter model on $\Lambda(N)$, the torus of side $N$ in $\mathbb{Z}^d$. Unlike $\eta_t, \eta^{(N)}_t$ (for fixed $N$) gets trapped with probability 1 as $t \rightarrow\infty$ at all 0's or all 1's. We examine the asymptotic growth of these trapping or consensus times $\tau^{(N)}$ as $N \rightarrow\infty$. To do this we obtain limit theorems for coalescing random walk systems on the torus $\Lambda(N)$, including a new hitting time limit theorem for (noncoalescing) random walk on the torus.

A measure on infinite self-avoiding walks is defined which is the natural limit of the uniform measure on finite self-avoiding walks. This limit is shown to exist in sufficiently large dimensions using the methods of Slade and Brydges and Spencer.

Let $X_1, X_2, X_3, \ldots$ be a sequence of iid $\mathbb{R}^2$-valued bounded random variables with mean vector zero and covariance matrix identity. Let $S = (S_n; n \geq 0)$ be the random walk defined by $S_n = \sum^n_{i = 1} X_i$. Let $\phi(n)$ be the winding of $S$ at time $n$, that is, the total angle wound by $S$ around the origin up to time $n$. Under a mild regularity condition on the distribution of $X_1$, we show that $2\phi(n)/\log n \rightarrow_d W$ where $\rightarrow_d$ denotes convergence in distribution and where $W$ has density $(1/2)\operatorname{sech}(\pi w/2)$.

We prove, in all dimensions, that for a stationary Gibbs state with finite range or rapidly decreasing interaction, there is at most one infinite percolation cluster. This implies that the connectivity function is bounded away from 0.

Gibbs measures are studied using a Markov chain on the nonnegative integers. Uniqueness of Gibbs measures follows from absorption of the chain at $\{0\}$. To this end, we derive a certain inequality. For one-dimensional systems this extends a well-known uniqueness result of Ruelle and for models near the $1/r^2$-interaction Ising model it is a natural improvement of some other results.

Consider a random field on $R^d, d \geq 1$. A simple condition is given on the covariance function which ensures the existence of a version of the random field in which the realizations are everywhere continuous. The proof involves a rather delicate approximation of the random field by interpolating polynomials of suitably high order.

Let $I_p(f)$ and $I_q(g)$ be multiple Wiener-Ito integrals of order $p$ and $q$, respectively. A characterization of independence of general random variables on Wiener space in the context of the stochastic calculus of variations is derived and a necessary and sufficient condition on the pair of kernels $(f, g)$ is derived under which the random variables $I_p(f), I_q(g)$ are independent.

We show that if $2N \leq d$, then with probability 1 the Brownian sheet $W: R^N_+ \rightarrow R^d$ satisfies $\forall$ Borel set $E, \dim W(E) = 2 \dim E$.

We show that the diameter of the support of a measure-valued critical branching Brownian motion tends to zero almost surely at the time of extinction.

We consider a multitype population size dependent branching process in discrete time. A process is considered to be near-supercritical if the mean matrices of offspring distributions approach the mean matrix of a supercritical process as the population size increases. We show that if the convergence of the means to the supercritical mean is fast enough and the second moments of offspring distributions do not grow too fast as the population size increases, then the process grows geometrically fast. Similarly to the classical multitype Galton-Watson process, the process grows at the geometric rate determined by the largest eigenvalue of the limiting matrix in the direction of the corresponding left eigenvector.

Let $\{P^{(k)}_t, t \geq 0\}, k = 1, 2$, be two Markov semigroups on $\hat{C}(E)$, the space of continuous functions on a separable, locally compact metric space $E$ which tend to zero at infinity. In this article, we derive a sufficient condition for the two semigroups to commute, in the sense that for each $s \geq 0, t \geq 0$ and each $f \in \hat{C}(E), P^{(1)}_s P^{(2)}_t f = P^{(2)}_t P^{(1)}_s f$.

Let $\{X(t): t \in R^+ \text{or} I^+\}$ be an (aperiodic) irreducible Markov process with a finite state space $S$ and transition rate $q_{ij}(t) = p(i, j)(\lambda(t))^{U(i, j)}$, where $0 \leq U(i, j) \leq \infty$ and $\lambda(t)$ is some suitable rate function with $\lim_{t \rightarrow \infty}\lambda(t) = 0$. We shall show in this article that there are constants $h(i) \geq 0$ and $\beta_i > 0$ such that independent of $X(0), \lim_{t \rightarrow \infty}P(X(t) = i) \div (\lambda(t))^{h(i)} = \beta_i$ for each $i \in S$. The height function $h$ is determined by $(p(i, j))$ and $(U(i, j))$. In particular, a limit distribution exists and concentrates on $\underline{S} = \{i \in S: h(i) = 0\}$.

We expand a multiple symmetric $\alpha$-stable integral $\int \cdots \int f(t_1, \ldots, t_n) dM(t_1) \ldots dM(t_n)$ into a LePage type multiple series of transformed arrival times of a Poisson process. An exact evaluation of the limit of appropriately normalized tail distribution results from this representation.

We define $\int^\bullet_0 X_s dY_s$ for $X$ a process locally of bounded $\beta$-variation and $Y$ locally of bounded $\alpha$-variation $(\alpha < 2 \leq \beta \text{and} 1/\alpha + 1/\beta > 1)$ as the limit of the Riemann sums. The properties of this integral lead us to an Ito formula and to the existence of local times for some kinds of Dirichlet processes.

It is shown that for a suitable definition of the nonadapted Stratonovich stochastic integral, the existence of the Ogawa integral implies that the Stratonovich integral exists and the two are equal.

A comparison theorem is proved for one-dimensional stochastic equations driven by continuous semimartingales and having Volterra-type drifts. A counterexample which shows that the coefficient of the continuous martingale term cannot be Volterra-type is given. Then the comparison result is used in order to obtain the existence of strong solutions when the Lipschitz condition is replaced by a Holder-type one.

We develop a new method to study the tails of a sum of independent mean zero Banach-space valued random variables $(X_i)_{i \leq N}.$ It relies on a new isoperimetric inequality for subsets of a product of probability spaces. In particular, we prove that for $p \geq 1,$ $\bigg\|\sum_{i \leq N} X_i\bigg\|_p \leq \frac{Kp}{1 + \log p}\bigg(\bigg\|\sum_{i \leq N} X_i\bigg\|_1 + \|\max_{i \leq N}\|X_i\|\|_p\bigg),$ where $K$ is a universal constant. Other optimal inequalities for exponential moments are obtained.

Using suitable self-normalizations for partial sums of i.i.d. random variables, a law of the iterated logarithm, which generalizes the classical LIL, is proved for all distributions in the Feller class. A special case of these results applies to any distribution in the domain of attraction of some stable law.

Expansions for the distribution of differentiable functionals of normalized sums of i.i.d. random vectors taking values in a separable Banach space are derived. Assuming that an $(r + 2)$th absolute moment exist, the CLT holds and the distribution of the $r$th derivative $r \geq 2$ of the functionals under the limiting Gaussian law admits a Lebesgue density which is sufficiently many times differentiable, expansions up to an order $O(n^{-r/2 + \varepsilon})$ hold. Applications to goodness-of-fit statistics, likelihood ratio statistics for discrete distribution families, bootstrapped confidence regions and functionals of the uniform empirical process are investigated.

We study tail $\sigma$-fields and loss of memory associated with sums of stationary integer-valued random variables. An application concerns convergence in distribution of interarrival times in zero-one sequences.

Bounds on the error in the normal approximation of sums of dependent random variables introduced by Stein are interpreted in terms of dependency graphs. This leads to improvements on a central limit theorem of Petrovskaya and Leontovich and recent applications by Baldi and Rinott. In particular, bounds on rates of convergence are obtained. As an application we study the normal approximation to the number of local maxima of a random function on a graph.

We show that for bounded domains $A \subseteq \mathbb{R}^N$ with $0\in A,$ if the exit time $\tau_A$ and exit place $X(\tau_A)$ are independent for a Brownian motion starting at 0, then $A$ is essentially a ball centered at 0. Extensions are given when $X(t)$ is a Brownian motion with constant drift and when $A$ is unbounded.

In this paper we consider the problem of coding a given stationary stochastic process to another with a prescribed marginal distribution. This problem after reformulation is solved by proving the following theorem. Let $(M, \mathscr{A}, \mu)$ be a Lebesgue probability space and let $\sigma$ be an antiperiodic bimeasurable $\mu$-preserving automorphism of $M.$ Let $\mathbf{N}$ be the set of nonnegative integers. Suppose that $(p_{i, j}: i, j \in \mathbf{N})$ are the transition probabilities of a positive recurrent, aperiodic, irreducible Markov chain with state space $\mathbf{N}$ and that $\pi = (\pi_i), i \in \mathbf{N},$ is the unique positive invariant distribution $\pi_j = \sum_{i \in \mathbf{N}}\pi_i p_{i, j}.$ Then there is a partition $\mathbf{P} = \{P_i\}_{i \in \mathbf{N}}$ of $M$ such that for all $i, j \in \mathbf{N}, \mu(P_i \cap \sigma^{-1}P_j) = \mu(P_i)p_{i, j} = \pi_ip_{i, j}.$

A contiguity dichotomy for two sequences of product measures is proved under the assumption of component measures belonging to a dominated experiment which is differentiable. This generalizes Eagleson's (1981) result for Gaussian measures. The dichotomy result is then used to generalize and clarify the results of Shepp (1965) and Steele (1986) with regards to finite Fisher information and equivalence dichotomies between two product measures, one with a fixed component measure and the second with rigidly perturbed component measures.

An exponential martingale is associated with the Markov chain of the number of customers in the $M/GI/1$ queue. With the help of arguments from renewal theory, this martingale provides a unified probabilistic framework for deriving several well-known generating functions for the $M/GI/1$ queue, such as the Pollaczek-Khintchine formula, the transient generating function of the number of customers at departure epochs and the generating function of the number of customers served in a busy period.