Nearest particle systems are continuous-time Markov processes on $\{0, 1\}^Z$ in which particles die at rate 1 and are born at rates which depend on their distances to the nearest particles to the right and left. There is a natural parametrization of these systems with respect to which they exhibit a phase transition. When the process is attractive and reversible, the critical value $\lambda_c$ above which a nontrivial invariant measure exists can be computed exactly. This invariant measure is the distribution $v$ of a stationary discrete time renewal process. Under a mild regularity assumption, we prove that the following three statements are equivalent: (a) The nearest particle system converges to equilibrium exponentially rapidly in $L_2(v)$. (b) The density of the interarrival times in the renewal process has exponentially decaying tails. (c) The nearest particle system is supercritical in the sense that $\lambda > \lambda_c$. Under an additional second-moment assumption, we prove that the critical exponent associated with the exponential convergence is 2. The proof of exponential convergence is based on an unusual comparison of the nearest particle system with an infinite system of independent birth and death chains. To carry out this comparison, a new representation is developed for a stationary renewal process with a log-convex renewal sequence in terms of a sequence of i.i.d. random variables.

Nearest-particle systems form a class of continuous-time interacting particle systems on $\mathbb{Z}$. The birth rate $\beta(l, r)$ at a given site depends on the distances $l$ and $r$ to the nearest occupied sites on the left and right; deaths occur at rate 1. Assume that $b(n) = \sum_{l + r = n} \beta(l, r), 2 \leq n < \infty, b(\infty) = \sum^\infty_{l =1} \beta(l, \infty) + \sum^\infty_{r=1} \beta(\infty, r)$, is constant. In Liggett [6] the question was posed whether for $b(n) \equiv 1 + \varepsilon, 2 \leq n \leq \infty$, with $0 < \varepsilon \leq 1$, there are such systems which survive for all $t$. Here, we answer affirmatively for all such $\varepsilon$ and construct a class of examples.

In this article we consider the asymptotic behavior of the contact process when the range $M$ goes to $\infty$. We show that if $\lambda$ is the total birth rate from an isolated particle, then the critical value $\lambda_c(M) \rightarrow 1$ as $M \rightarrow \infty$. The rate of convergence depends upon the dimension: $\lambda_c(M) - 1 \approx M^{-2/3}$ in $d = 1, \approx (\log M)/M^2$ in $d = 2$, and $\approx M^{-d}$ in $d \geq 3$.

The set of self-intersections of a Brownian path $b(t)$ taking values in $\mathbb{R}^3$ has Hausdorff dimension 1, for almost every such path, with respect to Wiener measure, a result due to Fristedt. Here we prove that this result (together with the corresponding result for paths in $\mathbb{R}^2$) in fact holds for quasi-every path with respect to the infinite-dimensional Ornstein-Uhlenbeck process, a diffusion process on Wiener space whose stationary measure is Wiener measure. We do this using Rosen's self-intersection local time, first proving that this exists for quasi-every path.

We prove a conjecture of Hendricks and Taylor that a Levy process in $\mathbb{R}^d$ with 1-potential kernel $u(x)$ will have $k$-multiple points if $\int_{|x| \leq 1} (u(x))^k dx < \infty$ and $u(0) > 0$.

Let $L(a,t)$ be the local time of a Wiener process. Our main result says that the process $L(a,t) - L(0,t)$ can be strongly approximated by a process obtained from a Wiener sheet $W(a,t)$ and a local time process $\hat{L}(0,t)$, independent of $W(\cdot,\cdot)$.

We study properties of functionals of the Brownian sheet: in particular, we construct square-integrable functionals of $R^d$-valued Brownian sheets, so that $k$-intersection local times on $R^{dk}$ can be obtained as specific examples. Results of central limit theorem type on sums of functionals of Brownian sheets, give us new fields, and we investigate which properties of the Brownian sheet (e.g., Markovianess, renormalizability) these limit fields inherit.

Let $L_\varepsilon = \varepsilon L_0 + L_1$, where $L_0$ is a nondegenerate elliptic operator on $R^2$ and $L_1 = \frac{1}{2}A(r, \theta)\partial^2/\partial\theta^2 + B(r, \theta)\partial/\partial\theta$. We assume that for fixed $r, L_1$ generates a positive recurrent diffusion on the circle with invariant measure $\mu_r(d\theta)$. Let $X^\varepsilon(t) = (r^\varepsilon(t), \theta^\varepsilon(t))$ denote the diffusion generated by $(1/\varepsilon)L_\varepsilon$ and let $u_\varepsilon$ be the solution to the Dirichlet problem $L_\varepsilon u = 0$ on $D$ and $u = f$ and $\partial D$, where $D = \{x: r_1 < |x| < r_2\}$ and $f$ is continuous. Then $u_\varepsilon(x) = E_x f(X^\varepsilon(\tau^\varepsilon_D))$, where $\tau^\varepsilon_D$ is the first exit time from $D$. By the averaging principle, the process $r^\varepsilon(t)$ converges weakly to the process $r^0(t)$ generated by $\bar{L}_0$, the operator obtained from $L_0$, by restricting to functions depending only on $r$ and averaging the coefficients with respect to $\mu_r(d\theta)$. Furthermore, $P_x(\theta^\varepsilon(\tau^\varepsilon_D \in d\theta\mid\tau^\varepsilon_{r_i} < \tau^\varepsilon_{r_j})$ converges weakly as $\varepsilon \rightarrow 0$, to a measure which can be calculated in terms of $\mu_{r_i}(\partial\theta)$ and the diffusion matrix of $L_0$, where $\tau^\varepsilon_r$ denotes the hitting time of the circle of radius $r$ and $(i, j) = (1, 2)$ or $(2, 1)$. The above information allows one to evaluate the limiting distribution of $(r^\varepsilon(\tau^\varepsilon_D), \theta^\varepsilon(\tau^\varepsilon_D))$ and thus also the asymptotics of $u_\varepsilon(x)$. Call $\theta$ the fast variable and $r$ the slow variable. In this paper we investigate what happens to the averaging principle in the case that the boundary of $D$ is no longer characteristic for the equation slow variable = constant.

Kesten (1969) gives necessary and sufficient conditions for single points to be polar for Levy processes. In this article we investigate when single points are polar for processes obtained by adding functions to stable processes.

To establish weak convergence of a sequence of martingales to a continuous martingale limit, it is sufficient (under the natural uniform integrability condition) to establish convergence of finite-dimensional distributions. Thus in many settings, weak convergence to a continuous limit process can be deduced almost immediately from convergence of finite-dimensional distributions. These results may be technically useful in simplifying proofs of weak convergence, particularly in infinite-dimensional settings. The results rely on a technical tightness condition involving stopping times and predictability of imminent jumps.

In this paper, we obtain several new results and developments in the study of empirical processes. A comparison theorem for Rademacher averages is at the basis of the first part of the results, with applications, in particular, to Kolmogorov's law of the iterated logarithm and Prokhorov's law of large numbers for empirical processes. We then study the behavior of empirical processes along a class of functions through random geometric conditions and complete in this way the characterization of the law of the iterated logarithm. Bracketing and local Lipschitz conditions provide illustrations of some of these ideas to concrete situations.

Let $X(t), 0 \leq t \leq 1$, be a real separable Gaussian process with mean 0 and continuous covariance function and put $\sigma^2(t) = EX^2(t)$. Under the well known conditions of Fernique and Dudley, the sample functions are continuous and there are explicit asymptotic upper bounds for the probability $P(\max_{\lbrack 0, 1 \rbrack}X(t) \geq u)$ for $u \rightarrow \infty$. Suppose that there is a point $\tau, 0 \leq \tau \leq 1$, such that $\sigma^2(t)$ has a unique maximum value at $t = \tau$ and put $\sigma = \sigma(\tau)$. The main result is a sharpening of the standard asymptotic upper bounds for $P(\max_{\lbrack 0, 1 \rbrack} X(t) \geq u)$ to take into account the existence of the unique maximum of $\sigma(t)$. Indeed, when the order of the standard bound exceeds that of the obvious lower bound $P(X(\tau) \geq u)$, the upper asymptotic bound is shown to be reducible by the factor $\int^1_0\exp(-u^2g(t)) dt$, with $g(t) = (1/\sigma)(1/\bar{\sigma}(t) - 1/\sigma)$, where $\bar{\sigma}(t)$ is an arbitrary majorant of $\sigma(t)$ satisfying certain general conditions. For a large class of processes the asymptotic order of the bound obtained in this way cannot be further reduced. The results are illustrated by applications to the ordinary Brownian motion and the Brownian bridge.

Several estimates of the rate of convergence of normalized maxima of random processes are exhibited by using the theory of probability metrics. In contrast to the summation scheme, uniform estimates for normalized maxima of random sequences will be derived.

We consider independent identically distributed observations taking values in a general partially ordered set. Under no more than a necessary measurability condition we develop a theory of record values analogous to parts of the well-known theory of real records, and discuss its application to many partially ordered topological spaces. In the particular case of $\mathbb{R}^2$ under a componentwise partial order, assuming the underlying distribution of the observations to be in the domain of attraction of an extremal law, we give a criterion for there to be infinitely many records.

Let $\{N(t), t \geq 0\}$ be the renewal process associated to an i.i.d. sequence $X_1, X_2, \ldots$ of nonnegative interarrival times having finite moment generating function near the origin. In this article we give strong and weak limiting laws for the maximal and minimal increments $\sup_{0 \leq t \leq T - K}(N(t + K) - N(t))$ and $\inf_{0 \leq t \leq T - k}(N(t + K) - N(t))$, where $K = K_T$ is a function of $T$ such that $0 \leq K_T \leq T$.

Let $R$ be the excess over the boundary in renewal theory. It is well known that $ER$ has a limit $r$ when the drift of the random walk $\mu \geq 0$. We study renewal theorems with varying $\mu$. Conditions are given under which the tail $ER - r$ is uniformly dominated by a decreasing integrable function for $\mu$ in a compact interval in $(0, \infty)$. Conditions are also given under which the derivative of the tail $(\partial/\partial\mu)(ER - r)$ is uniformly dominated by a directly Riemann integrable function.

Let $S_n = X_1 + \cdots + X_n$, where $X_1, X_2, \ldots$ are iid Banach-space-valued random variables with weak mean 0 and weak second moments. Let $K$ be the unit ball of the reproducing kernel Hilbert space associated to the covariance of $X$. We show that the cluster set of $\{S_n/(2n \log \log n)^{1/2}\}$ either is empty or has the form $\alpha K$, where $0 \leq \alpha \leq 1$. A series condition is given which determines the value of $\alpha$. In a companion paper, examples are given to show that all $\alpha \in \lbrack 0, 1 \rbrack$ do occur.

Given independent identically distributed random variables $\{X, X_{\bar{n}}; \bar{n} \in \mathbb{N}^d\}$ indexed by $d$-tuples of positive integers and taking values in a separable Banach space $B$ we approximate the rectangular sums $\{\sum_{\bar{k}} \leq \bar{n} X_{\bar{k}}; \bar{n} \in \mathbb{N}^d\}$ by a Brownian sheet and obtain necessary and sufficient conditions for $X$ to satisfy, respectively, the bounded, compact and functional law of the iterated logarithm when $d \geq 2$. These results improve, in particular, the previous work by Morrow [17].

Let $(B, \| \|)$ be a separable Banach space and $\mathscr{X} := \mathscr{X}(B)$ the vector space of all random variables defined on a probability space $(\Omega, \mathscr{A}, P)$ and taking values in $B$. It is shown that new ideal metrics for $\mathscr{X}$ may be used to obtain refined rates of convergence of normalized sums to a stable limit law. The rates hold uniformly in $n$ and are expressed in terms of a variety of uniform metrics on $\mathscr{X}$. In the Banach space setting the rates hold with respect to the total variation metric and in the Euclidean space setting the rates hold with respect to uniform metrics between density and characteristic functions. The main result provides a sharp order estimate of the rate of convergence in local limit theorems with respect to the uniform distance between densities. The method is based on the theory of probability metrics, especially those of convolution type.

Let $X$ be a quasinormed rearrangement invariant function space on (0, 1) which contains $L_q(0, 1)$ for some finite $q$. There is an extension of $X$ to a quasinormed rearrangement invariant function space $Y$ on $(0, \infty)$ so that for any sequence $(f_i)^\infty_{i = 1}$ of symmetric random variables on (0,1), the quasinorm of $\sum f_i$ in $X$ is equivalent to the quasinorm of $\sum\mathbf{f}_i$ in $Y$, where $(\mathbf{f}_i)^\infty_{i = 1}$ is a sequence of disjoint functions on $(0, \infty)$ such that for each $i, \mathbf{f}_i$ has the same decreasing rearrangement as $f_i$. When specialized to the case $X = L_q(0, 1)$, this result gives new information on the quantitative local structure of $L_q$.

Suppose $\{X_n\}$ is an i.i.d. sequence of random variables with mean 0, variance 1 and $S_n = \sum^n_{i = 1}X_i$. Let $0 < r < 1$. It is well known that $S_n - W(n) = O((\log n)^{1/r}) \mathrm{a.s}.$ when $Ee^{t_0|X_1|^r} < \infty$ for some $t_0 > 0$, where $\{W(t), t \geq 0\}$ is the standard Wiener process. We prove that $O((\log n)^{1/r})$ cannot be replaced by $o((\log n)^{1/r})$.