We give a new type of characterization of the Glivenko-Cantelli classes. In the case of a class $\mathscr{L}$ of sets, the characterization is closely related to the configuration that the sets of $\mathscr{L}$ can have. It allows one to decide simply whether a given class is a Glivenko-Cantelli class. The characterization is based on a new measure theoretic analysis of sets of measurable functions. This analysis also gives an approximation theorem for Glivenko-Cantelli classes, sharpenings of the Vapnik-Cervonenkis criteria and the value of the asymptotic discrepancy for classes that are not Glivenko-Cantelli. An application is given to the law of large numbers in a Banach space for functions that need not be random variables.

A refinement of the Komlos, Major and Tusnady (1975) inequality for the supremum distance between the uniform empirical process and a constructed sequence of Brownian bridges is obtained. This inequality leads to a weighted approximation of the uniform empirical and quantile processes by a sequence of Brownian bridges dual to that recently given by M. Csorgo, S. Csorgo, Horvath and Mason (1986). The present theory approximates the uniform empirical process more closely than the uniform quantile process, whereas the former theory more closely approximates the uniform quantile process.

Let $X_k$ be i.i.d., $S_n = X_1 + \cdots + X_n$, and $X^{(1)}_n$ the term of maximum modulus among $\{X_1,\ldots, X_n\}$. Let $u_k = P\{2^k < |X_1| \leq 2^{k+1}\parallel |X_1| > 2^k\}$. The main result is that $X^{(1)}_n/S_n \rightarrow 1$ a.s. $\operatorname{iff} \sum u^2_k < \infty$. Furthermore, for any positive integer $r, \lim\inf_{n\rightarrow\infty} |X^{(1)}_n/S_n| = r^{-1} \mathrm{a.s.} \operatorname{iff} \sum_k u^r_k = \infty$ and $\sum_ku^{r+1}_k < \infty$. If $\sum_ku^r_k = \infty$ for all $r$ then $\lim\inf_{n\rightarrow\infty} |X^{(1)}_n/S_n| = 0$ a.s.

Let $(\mathbf{S}, \rho)$ be a metric space, $(\mathbf{V}, \mathscr{V}, \mu)$ be a probability space, and $f: \mathbf{S} \times \mathbf{V} \rightarrow \mathbb{R}$ be a real-valued function on $\mathbf{S} \times \mathbf{V}$ which has mean zero and is Lipschitz in $L_2(\mu)$ with respect to $\rho$. Let $V$ be a random variable defined on $(\mathbf{V}, \mathscr{V}, \mu)$, and let $\{V_i: i \geq 1\}$ be a sequence of independent copies of $V$. The limiting behavior of the process $S_n(s) = n^{-1/2}\sum^n_{i=1} f(s, V_i)$ is studied under an integrability condition on the metric entropy with bracketing in $L_2(\mu)$. This metric entropy condition is analogous to one which implies the continuity of the limiting Gaussian process. A tightness result is derived which, in conjunction with the results of Andersen and Dobric (1987), shows that a central limit theorem holds for the $S_n$-process. This result generalizes those of Dudley (1978), Dudley (1981) and Jain and Marcus (1975).

Recent advances in the theory of Edgeworth expansion allow rigorous estimation of remainder terms, under explicit regularity conditions. However, even in the important case of Student's $t$ statistic, the moment conditions required are unrealistically stringent. In this note we relax those conditions to the minimum needed to define terms in the Edgeworth expansion.

Let $X_1, X_2,\ldots$ be independent observations from a distribution with a regularly varying upper tail with index $a$ greater than 2. For each $n \geq 1$, let $X_{1,n} \leq \cdots \leq X_{n,n}$ denote the order statistics based on $X_1,\ldots, X_n$. Choose any sequence of integers $(k_n)_{n\geq 1}$ such that $1 \leq k_n \leq n, k_n \rightarrow \infty$, and $k_n/n \rightarrow 0$. It has been recently shown by S. Csorgo and Mason (1986) that the sum of the extreme values $X_{n,n} + \cdots + X_{n-k_n,n}$, when properly centered and normalized, converges in distribution to a standard normal random variable. In this paper, we completely characterize such sequences $(k_n)_{n\geq 1}$ for which the corresponding law of the iterated logarithm holds.

We give conditions for transience and recurrence for certain diffusion processes in the plane which "look" recurrent in certain sectors and transient in others. We also give conditions for the convergence of diffusion paths to paths of the deterministic dynamical system corresponding to the first-order part of $L$, the generator of the process. This in turn is related to the question of existence of bounded harmonic functions for the operator $L$. Conditions are given for the existence and nonexistence of bounded harmonic functions for $L$.

Let $X_t \in \mathbf{R}^d$ be the solution to the stochastic differential equation $dX_t = \sigma(X_t) dB_t + b(X_t) dt, X_0 \in \mathbf{R}^d,$ where $B_t$ is a Brownian motion in $\mathbf{R}^d$. The aim of this paper is to make the following statement precise: "Let $x_t$ be a solution of $\dot{x} = b(x)$. If $|x_t| \rightarrow \infty$ as $t \rightarrow \infty$ and the drift vector field $b(x)$ is well behaved near $x_t$ then with positive probability, $X_t \rightarrow \infty$, and does so asymptotically like $x_t$." Examples are provided to illustrate the situations in which this theorem may be applied.

For a simply connected plane Greenian domain $D$ we give a natural description of the means of exit of a Brownian path from $D$. We give a related discussion of the connection between classical and probabilistic Fatou theorems.

Ventsel and Freidlin studied small random perturbations $dx^\varepsilon(t) = b(x^\varepsilon(t)) dt + \sqrt{\varepsilon} \sigma (x^\varepsilon(t)) d\beta(t)$ of the dynamical system $dx(t) = b(x(t)) dt$. They proved a large deviations theorem for the family $\{x^\varepsilon(t)\}$, and used the large deviations estimates to prove (among other things) a result about exit from a bounded domain containing an equilibrium of the unperturbed system. Fleming rederived this exit theorem using techniques from stochastic control theory. In this paper we study analogous questions for singular perturbations of degenerate diffusions. We consider the family of processes $dx^\varepsilon(t) = b(x^\varepsilon(t)) dt + \tau(x^\varepsilon(t))\circ dz(t) + \sqrt{\varepsilon} \sigma(x^\varepsilon(t)) d\beta(t)$ for $\varepsilon \geq 0$, where the process $x^0(t)$ is degenerate. We show that a large deviations principle need not hold, but we derive bounds sufficient for obtaining some estimates on probabilities. We also adapt Fleming's approach to prove an exit theorem.

Let $X^x_\alpha$ be a Bessel process with parameter $\alpha$, starting at $x \geq 0$. Gordon [3] obtained $L^p$ inequalities which relate stopping times to stopping places for the case $\alpha = 1, x = 0$ and $p > \frac{1}{2}$. Rosenkrantz and Sawyer [5] extended them to $\alpha > 0, x = 0$ and $p \geq 1$. Burkholder [1] obtained results for $\alpha$ a positive integer, $x \geq 0$ and $p > 0$. Here we consider arbitrary starting points $x, \alpha > 0$ and $p > 0$. The $L^p$ inequalities are valid for $\alpha \geq 2$ with $p > 0$, and also for $0 < \alpha < 2$ with $p > (2 - \alpha)/2$. Examples are constructed to show that for $0 < \alpha < 2$ with $p \leq (2 - \alpha)/2$, the $L^p$ inequalities cannot hold.

We prove a weak limit theorem which relates the large time behavior of the maximum of a branching Brownian motion to the limiting value of a certain associated martingale. This exhibits the minimal velocity travelling wave for the KPP-Fisher equation as a translation mixture of extreme-value distributions. We also show that every particle in a branching Brownian motion has a descendant at the frontier at some time. A final section states several conjectures concerning a hypothesized stationary "standing wave of particles" process and the relationship of this process to branching Brownian motion.

Let $X, X_1,\ldots, X_d$ be a sequence of independent, symmetric, identically distributed random vectors with independent components. The main subject of this paper is the so-called decoupling inequalities, i.e., inequalities of the form \begin{align*}E\phi (cQ(X, X,\ldots, X)) &\leq E\phi (Q(X_1, X_2,\ldots, X_d)) \\ &\leq E\phi(CQ(X, X,\ldots, X)), \\ \end{align*} where $Q$ is a symmetric multilinear form with values in a vector space $F$ with all "diagonal" terms equal to zero and $\phi$ is a convex function on $F$.

Let $X(t), t \geq 0$, be a process with independent, symmetric and stationary increments and let $(\xi_i)$ be i.i.d. symmetric real random variables. We provide a characterization of functions $f(s, t), s, t \geq 0$, such that the double integral $\int\int f(s, t) dX(s) dX(t)$ exists, a characterization of infinite matrices $(\alpha_{ij})$ such that the double series $\sum\alpha_{ij}\xi_i\xi_j$ converges a.s. and a characterization of Orlicz space $l_\psi$ valued sequences $(a_i)$ for which the series $\sum a_i\xi_i$ converges a.s. in $l_\psi$. The above three problems are closely related.

Let $M = (M_t)_{t\in\lbrack 0,1\rbrack^2}$ be a two-parameter $L\log^+ L$-bounded (not necessarily continuous) martingale. Assume that the marginal filtrations in the first and second directions are quasi-left continuous. We prove the existence of quadratic variation in the sense of convergence in probability. This is done first for bounded martingales. The extension to the general case is obtained by approximating a given martingale by its bounded truncations and using a two-parameter version of the square function inequality of Burkholder.

Consider a binary Markov random field whose neighbor structure is specified by a countable graph with nodes of uniformly bounded degree. Under a minimal assumption we prove a decomposition theorem to the effect that such a Markov random field can be represented as the nodewise modulo 2 sum of two independent binary random fields, one of which is white binary noise of positive weight. Said decomposition provides the information theorist with an exact expression for the per-site rate-distortion function of the random field over an interval of distortions not exceeding this weight. We mention possible implications for communication theory, probability theory and statistical physics.

Let $\delta_c(X)$ denote the maximum $d$ in $\lbrack 0, \frac{1}{2}\rbrack$ such that a binary Gibbs random field $X$ can be decomposed as the modulo 2 sum of two independent binary fields, one of which is independent Bernoulli (white binary noise) of weight $d$. In a recent paper, Hajek and Berger showed, under modest assumptions, that $\delta_c > 0$. We point out here that the decomposition of $X$ is related to the classic statistical mechanics problem of determining zero-free regions of partition functions. A theorem of Ruelle is then applied to obtain improved estimates for $\delta_c$.

We consider the one dimensional supercritical contact process with initial configurations having infinitely many particles to the left of the origin and only finitely many to its right. Starting from any such configuration, we first prove that in the limit as time goes to infinity the law of the process, as seen from the edge, converges to the invariant distribution constructed by Durrett [12]. We then prove a functional central limit theorem for the fluctuations of the edge around its average, showing that the corresponding diffusion coefficient is strictly positive. We finally characterize the space time structure of the system. In particular we prove that its distribution shifted in space by $\alpha t$ ($t$ denotes the time and $\alpha$ the drift of the edge) converges when $t$ goes to infinity to a $\frac{1}{2} - \frac{1}{2}$ mixture of the two extremal invariant measures for the contact process.

We prove that the edge process of oriented percolation in two dimensions does not have a stationary distribution in the subcritical case.

Random number generators often work by recursively computing $X_{n+1} \equiv aX_n + b (\mod p)$. Various schemes exist for combining these random number generators. In one scheme, $a$ and $b$ are themselves chosen each time from another generator. Assuming that this second source is truly random, we investigate how long it takes for $X_n$ to become random. For example, if $a = 1$ and $b = 0, 1$, or $-1$ each with probability $\frac{1}{3}$, then $cp^2$ steps are required to achieve randomness. On the other hand, if $a = 2$ and $b = 0, 1$, or $-1$, each with probability $\frac{1}{3}$, then $c \log p \log\log p$ steps always suffice to guarantee randomness, and for infinitely many $p$, are necessary as well, although, in fact, for almost all odd $p, 1.02 \log_2 p$ steps are enough.

This paper is concerned with the stationary distribution of a one-dimensional circuit-switched network. We show that if arrival rates decay geometrically with distance, then under the stationary distribution the number of circuits busy on successive links of the network at a fixed point in time is a Markov chain. When each link of the network has unit capacity we show that translation invariant arrival rates lead to a stationary distribution which can be described in terms of an alternating renewal process.

An allocation model [$n$ balls, $m (\geq n)$ cells, at most one ball in a cell] related to a hashing algorithm is studied. A ball $x$ goes into the cell $h(x)$, where $h(\cdot): \{1,\cdots, n\} \rightarrow \{1, \cdots, m\}$ is random. In case the cell $h(x)$ is already occupied, the ball $x$ is rejected and moved into the leftmost empty cell. This empty cell is found via the sequential search from left to right starting with the cell occupied by the last (before $x$) rejected ball. Denote $T_2(x)$ the number of the necessary probes. In the end, due to a resulting system of references, the $n$ occupied cells form a disjoint union of ordered chains, and to locate a ball $x$ it suffices to search only the cells of a subchain originating at the cell $h(x)$. Denote $T_1(x)$ the length of this subchain. The main result of the paper is: in probability, $\max T_1(x) = \log_bn - 2\log_b\log n + O(1),$ $\max T_2(x) = \log_bn - \log_b\log n + O(1),$ as $n \rightarrow \infty$, if $n/m$ is bounded away from $0, b = (1 - e^{-n/m})^{-1}$.

We prove that a Gaussian Taylor series takes almost surely every complex value with at most one exception. This presents an answer to a problem of Kahane. We also present an example to show some differences between Gaussian and classical Taylor series.

Using a new approach, an extended version of Spitzer's integral representation for stationary measures of a discrete branching process is obtained. This result is used to provide a complete solution to a problem in damage models satisfying a generalized Rao-Rubin condition.

The preorder relation of Hardy, Littlewood and Polya (1929), Day (1973) and Chong (1974, 1976) is applied to multivariate probability densities. This preorder, which is called majorization here, can be interpreted as an ordering of randomness. When used to compare multivariate densities with the same marginal densities, it can be interpreted as an ordering of dependence or conditional dependence. Results in Hickey (1983, 1984) and Joe (1985) are generalized. A relative entropy function is proposed as a measure of dependence or conditional dependence for multivariate densities with the same marginals.