In this paper we introduce a percolation construction which allows us to reduce problems about supercritical contact processes to problems about 1-dependent oriented percolation with density $p$ close to 1. Using this method we obtain a number of results about the growth of supercritical contact processes and the wet region in oriented percolation. As a corollary to our results we find that the critical probability for oriented site percolation is greater than (>) that for bond percolation.

We consider certain Markov processes with state space $\{0, 1\}^z$ which were introduced and first studied by Spitzer. In these systems, deaths occur at rate one independently of the configuration, and births occur at rate $\beta(\ell, r)$ where $\ell$ and $r$ are the distances to the nearest particles to the left and right, respectively. In his paper, Spitzer gave a necessary and sufficient condition for this process to have a reversible invariant measure, and showed that such a measure must be a stationary renewal process. It was that fact which motivated the study of these systems. Assuming that the process is attractive in the sense of Holley, we give conditions under which (a) the pointmass on the configuration "all zeros" is invariant, and (b) the reversible renewal process is the only nontrivial invariant measure which is translation invariant. As an application, these results allow us to determine exactly the values of $\lambda > 0$ and $p > 0$ for which the process with $\beta(\ell, r) = \lambda(1/\ell + 1/r)^p$ is ergodic.

Upper and lower bounds are obtained for $P(|W(t)| \leq f(t), t \leq u)$ and $P(|S(n)| \leq f(n), n \leq N), u, N$ large, where $W(t)$ is a Brownian motion, $S(n)$ is a random walk with $ES(1) = 0, E|S(1)|^{2+2\eta} < \infty$, and $f(t)$ is a deterministic function growing faster than $t^{1/2}$ but slower than $(2t \ln \ln t)^{1/2}$.

For a random walk with mean zero and variance one, a conditional limit theorem is proved under conditions on the path until it for the first time becomes negative. This gives a generalization of a limit theorem for random walk conditioned to stay positive which was considered by Iglehart and others. It has an application to get a tail formula of the d.f. of the maximum for a stopped random walk.

For $p \geq 1$, conditions for a separable Gaussian process to have sample paths of finite $p$-variation are given in terms of the mean function and the covariance function. A process with paths of finite $p$-variation may or may not induce a tight measure on the nonseparable Banach space $B_p$. Consequences of tightness and conditions for tightness are given.

Let $M$ be a symmetric independently scattered random measure on $\lbrack 0, 1\rbrack$ with control measure $m$ which is uniformly in the domain of normal attraction of a stable measure of index $p \in (0, 2\rbrack$. Let $f$ be a non-anticipating process with respect to $X(t) = M\lbrack 0, t\rbrack$ if $m$ is continuous, and a previsible process in general, satisfying $\int^1_0 E|f|^p dm < \infty$. Then the stochastic integral $\int^t_0 f dM$ can be defined as a process in $D\lbrack 0, 1\rbrack$ and is in the domain of normal attraction of a stable process of order $p$ in $D\lbrack 0, 1\rbrack$ in the sense of of weak convergence of probability measures. If $M$ is Gaussian and continuous in probability then the central limit theorem holds in $C\lbrack 0, 1\rbrack$; in particular, Ito and diffusion processes satisfy the CLT. Our main tool is an upper bound for the weak $L^p$ norm of $\sup_{0 \leq t \leq 1} |\int^t_0 f dM|$ in terms of the $L^p(P \times m)$ norm of $f$.

We prove a small deviation theorem of a new form for the functional central limit theorem for partial sums of independent, identically distributed finite-dimensional random vectors. The result is applied to obtain a functional form of the Chung-Jain-Pruitt law of the iterated logarithm which is also a strong speed of convergence theorem refining Strassen's invariance principle.

We investigate the cluster set $C(\{S_n/a_n\})$ under conditions necessary for the bounded law of the iterated logarithm, and obtain necessary and sufficient conditions for the LIL in spaces satisfying a certain comparison principle. In particular, these results settle some previously unanswered questions in the Hilbert space setting.

Let $X_1, X_2, \cdots$ be independent random variables with common distribution function $F$, zero mean, unit variance, and finite moment generating function, and with partial sums $S_n$. According to the strong law of large numbers, $p_m \equiv P\big\{\frac{S_n}{n} > c_n \text{for some} n \geq m\big\}$ decreases to 0 as $m$ increases to $\infty$ when $c_n \equiv c > 0$. For general $c_n$'s the Hewitt-Savage zero-one law implies that either $p_m = 1$ for every $m$ or else $p_m \downarrow 0$ as $m \uparrow \infty$. Assuming the latter case, we consider here the problem of determining $p_m$ up to asymptotic equivalence. For constant $c_n$'s the problem was solved by Siegmund (1975); in his case the rate of decrease depends heavily on $F$. In contrast, Strassen's (1967) solution for smoothly varying $c_n = o(n^{-2/5})$ is independent of $F$. We complete the solution to the convergence rate problem by considering $c_n$'s intermediate to those of Siegmund and Strassen. The rate (Theorem 1.1) in this case depends on an ever increasing number of terms in the Cramer series for $F$ the more slowly $c_n$ converges to zero.

Let $\mu$ be a probability measure and $H(x) = \sum^\infty_{n=0} \mu^{n_\ast}(-\infty, x\rbrack$ its renewal function. It is well-known that $H(x) - x/\mu_1 - \mu_2/2\mu^2_1 \rightarrow 0$ as $x \rightarrow +\infty$ if $\mu_1 > 0$ and $\mu$ is a nonlattice measure. ($\mu_k$ is the $k$th moment of $\mu$.) The rate of this convergence is studied under further conditions on $\mu$.

Let $\mu(\cdot)$ be a probability measure on $\mathbb{R}^d$ and $B$ be a convex set with nonempty interior. It is shown that there exists a unique "dominating" point associated with $(\mu, B)$. This fact leads (via conjugate distributions) to a representation formula from which sharp asymptotic estimates of the large deviation probabilities $\mu^{\ast n}(nB)$ can be derived.

The univariate conditions of Gnedenko characterizing domains of attraction for univariate extreme value distributions are generalized to higher dimensions. In addition, it is shown that random variables with a multivariate extreme value distribution are associated. Applications are given to a number of parametric families of joint distributions with given marginal distributions.

The idea of the domain of normal attraction was earlier extended to probabilities on a finite-dimensional inner-product space. We obtain a necessary and sufficient condition that a probability be in the domain of normal attraction of a given probability in terms of their covariance operators and of a limit involving the Levy measure. This condition appears to be the natural generalization of the corresponding univariate condition. We also show that the domains of normal attraction of two probabilities are either the same or disjoint, with a condition that is necessary and sufficient for them to be the same.

We study the limiting behavior of sums of functions of nearest neighbor distances for an $m$ dimensional sample. We establish a central limit theorem and moment bounds for such sums and an invariance principle for the empirical process of nearest neighbor distances. As a consequence we obtain the asymptotic behavior of a practicable goodness of fit test based on nearest neighbor distances.

The determinant of the fundamental matrix of a linear system of stochastic differential equations is computed.

In this paper we consider specific directed graphs called "ladders". The vertices of the graph are randomly colored by green or red. Deleting the edges with at least one red endpoint one gets a random graph. We give a method for finding the exact asymptotics of the longest path of this random graph if the "height" of the ladder goes to infinity. The result is a generalization of a celebrated theorem of Erdos-Renyi. An example is given, illustrating the method.

In this paper we prove a strong law of large numbers for random sets whose values are compact convex subsets of a Banach space.

By means of a time reversal argument based on Nagasawa's lemma, the equivalence of two representations for the Brownian local time is demonstrated.