In 1963 Glauber analyzed a one dimensional model for magnetism. It was the first study of the Markovian time evolution of a system with infinitely many interacting components. In Section 1 it will be discussed in the light of recent progress in this field. The remaining Sections (2, 3, and 4) form an introduction to recent joint work with Thomas M. Liggett. They concern new types of systems where each component takes on a real value which fluctuates in a way depending linearly on the values of neighboring components. The ergodic theory for such systems with finitely many components is the subject of Section 2. The results suggest conjectures for the case of infinitely many components, stated in Section 3 and proved in the joint paper with T. Liggett (ibid.). Section 4 introduces another class of time evolutions whose ergodic behavior may be analyzed by similar methods.

A limit theorem is established for a class of random processes (called here subadditive Euclidean functionals) which arise in problems of geometric probability. Particular examples include the length of shortest path through a random sample, the length of a rectilinear Steiner tree spanned by a sample, and the length of a minimal matching. Also, a uniform convergence theorem is proved which is needed in Karp's probabilistic algorithm for the traveling salesman problem.

We investigate the relationship between the bounded law of the iterated logarithm and the compact law of the iterated logarithm for Banach space valued random variables through the cluster set of $S_n/a_n$. Some rate of escape questions are also examined.

We obtain a nonuniform estimate of the rate of convergence in the martingale central limit theorem for convergence to mixtures of normal distributions. The uniform rates of convergence obtained by several other authors are special cases of our nonuniform estimate. We also obtain a rate of convergence in B. M. Brown's central limit theorem, assuming only Brown's elementary conditions. This result is a martingale analogue of Feller's generalization of the Berry-Esseen theorem.

The averages of an identically distributed martingale difference sequence converge in mean to zero, but the almost sure convergence of the averages characterizes $L \log L$ in the following sense: if the terms of an identically distributed martingale difference sequence are in $L \log L$, the averages converge to zero almost surely; but if $f$ is any integrable random variable with zero expectation which is not in $L \log L$, there is a martingale difference sequence whose terms have the same distributions as $f$ and whose averages diverge almost surely. The maximal function of the averages of an identically distributed martingale difference sequence is integrable if its terms are in $L \log L$; the converse is false.

Let $X_1, X_2, \cdots$ be a sequence of independent random variables with $S_n = \sum^n_{i = 1} X_i$. Fix $\alpha > 0$. Let $\Phi(\cdot)$ be a continuous, strictly increasing function on $\lbrack 0, \infty)$ such that $\Phi(0) = 0$ and $\Phi(cx) \leq c^\alpha\Phi(x)$ for all $x > 0$ and all $c \geq 2$. Suppose $a$ is a real number and $J$ is a finite nonempty subset of the positive integers. In this paper we are interested in approximating $E \max_{j \in J} \Phi(|a + S_j|)$. We construct a number $b_J(a)$ from the one-dimensional distributions of the $X$'s such that the ratio $E \max_{j \in J} \Phi(|a + S_j|)/\Phi(b_J(a))$ is bounded above and below by positive constants which depend only on $\alpha$. Bounds for these constants are given.

By transforming a central limit theorem for dependent variables, we find conditions for a sequence of processes with paths in $D\lbrack 0, \infty)$ to converge weakly to a diffusion process. Of the most important conditions, the first is related to (but weaker than) tightness, and in the next two we require that the first two conditional moments, given the past, of truncated increments in small time intervals, should stay close to the appropriate infinitesimal coefficients of the limiting diffusion times the length of the time interval. The limiting diffusions can have inaccessible or exit boundaries. We prove that our conditions are necessary and sufficient in order that: (1) the sequence of processes converges weakly in $D\lbrack 0, \infty)$; (2) any finite number of conditional expectations given the past of bounded, continuous functionals of the processes converge jointly in distribution to the "correct" value.

A dichotomy for sequences of pairs of Gaussian measures is proved. This result is then used to give a simple proof of the famous equivalence/singularity dichotomy for Gaussian processes. The proof uses tightness arguments and can be directly applied to the theory of hypothesis testing to show that two sequences of simple hypotheses which specify Gaussian measures are either contiguous or entirely separable.

The main result establishes that row sums $S_n$ of a double array of rowwise independent, infinitesimal (or merely uniformly asymptotically constant) random variables satisfying $\lim \sup |S_n - M_n| \leq M_0 < \infty$ a.c. (for some choice of constants $M_n$), obey a weak law of large numbers, i.e., $S_n - \operatorname{med} S_n$ converges in probability to 0. No moment assumptions are imposed on the individual summands and zero-one laws are unavailable. As special cases, a new result for weighted i.i.d. random variables and a result of Kesten are obtained.

Let $F$ be an infinitely divisible distribution on $\lbrack 0, \infty)$, with Levy measure $\nu$. For all real $r$, define measures $\nu_r$ by $\nu_r(dx) = x^r\nu(dx) (x > 1), = 0 (x \leq 1)$. For $0 < \alpha < \infty$, and $- \infty < r' < \alpha < r < \infty$, it is proved that $\nu_{r'}(x, \infty)$ is regularly varying (at $\infty$) with exponent $r' - \alpha$ if and only if $1 - F$ is regularly varying with exponent $- \alpha$ if and only if $\nu_r(0, x\rbrack$ is regularly varying with exponent $r - \alpha.$ If any of this is the case there follow asymptotic relations between $1 - F$ and either of $\nu_{r'}(x, \infty)$ or $\nu_r(0, x\rbrack$. The paper characterises those distributions for which these asymptotic relations hold, some of the characterisations being complete and others assuming that not all moments of $F$ are finite. The characterising classes involve regular variation, second order (de Haan) regular variation, rapid variation, and subexponentiality. An intermediate result is that when $F$ has finite $n$th and infinite $(n + 1)$th moment, $\int^t_0 x^{n + 1}\{1 - F(x)\} dx \sim \int^t_0 x^{n + 1}\nu(x, \infty) dx$ as $t \rightarrow \infty$. The results are applied to generalised gamma convolutions.

Operator-stable distributions are the $n$-dimensional analogues of stable distributions when nonsingular matrices are used for scaling. Every full operator-stable distribution $\mu$ has an exponent, that is, a nonsingular linear transformation $A$ such that for every $t > 0 \mu^t = \mu t^{-A}\ast\delta(a(t))$ for some function $a: (0, \infty) \rightarrow R^n$. Full operator-stable distributions on $R^2$ have multiple exponents if and only if they are elliptically symmetric; in this case the characteristic functions are of the form $\exp\{iy'Vw - c|Vy|^\gamma\}$ where $V$ is positive-definite and self-adjoint, $0, < \gamma \leq 2, c > 0$, and $w$ is a point in $R^2$.

The critical age-dependent branching process with offspring p.g.f. of the form $f(s) = s + (1 - s)^{1 + \alpha}L(1 - s), 0 < \alpha \leq 1, L$ slowly varying at 0, is investigated. We generalize Kesten's unpublished necessary condition to establish N.A.S.C. on the tail of the lifetime distribution for existence of a nondegenerate normalized conditioned limit law and pose several related questions.

It is shown that the well-known problem of determining the probability of extinction in a simple branching process has a duality relation to the problem of determining that offspring distribution which is in a sense closest to the original one and for which the new process is subcritical (or critical). The latter problem is also considered with respect to various measures of distance.

A simple proof of Wiener's multiparameter ergodic theorem is given in this paper. At the same time it is shown that two of the hypotheses of an existing proof can be dispensed with.

Let $\xi_0, \xi_1, \cdots, \xi_n, \cdots$ be a Gaussian stationary sequence of random variables. We study the asymptotic behavior of the mean number of real roots of the polynomial $P_n(x) = \xi_0 + \xi_1x + \cdots + \xi_nx^n$ as $n \rightarrow \infty$.

Suppose $X_n$ is an i.i.d. sequence of random variables with mean $\mu$ and that $t_n$ is a nondecreasing sequence of positive integers such that $t_n \leq n$. Let $S_n = X_1 + \cdots + X_n$. We give conditions under which $\max_{t_n \leq k \leq n} \big|\frac{S_n - S_{n - k}}{k} - \mu \big| \rightarrow 0$ almost surely and we discuss sharpness.

A strictly increasing, pure jump process with stationary, independent increments hits a single point $r > 0$ with probability 0. Adapting a method of proof, due to Carleson, we obtain a similar result for processes with exchangeable increments. This enables us to solve a regularity problem from game theory concerning probabilities of covering single points by randomly ordered intervals.

Let $X(t)$ be a homogeneous process with independent increments having the representation $X(t) = W(t) + \int_{x \neq 0} x\nu^\ast(t, dx)$, where $W(t)$ is a Wiener process with parameter $\sigma^2$ and $\nu^\ast(t, dx) = v(t, dx) - t\mu(dx)$, where $\nu(t, dx)$ is a Poisson random measure with mean measure $t\mu(dx)$. If the $m$th absolute mean of $X(t)$ is finite, then $\int^t_0 dX(t_1) \int^{t_1}_0 dX(t_2) \cdots \int^{t_m - 1}_0 dX(t_m) = \{\partial^m/\partial u^m \exp\{uW(t) + \int_{x \neq 0} \log(1 + ux)\nu^\ast(t, dx) - 1/2tu^2\sigma^2 - t \int_{x \neq 0} \lbrack ux - \log(1 + ux)\rbrack\mu(dx)\}\}_{u = 0}/m!$.

The following inequality is useful in studying a variation of the classical isoperimetric problem. Let $X$ be normally distributed with mean 0 and variance 1. If $g$ is absolutely continuous and $g(X)$ has finite variance, then $E \{\lbrack g'(X)\rbrack^2\} \geq \operatorname{Var}\lbrack g(X)\rbrack$ with equality if and only if $g(X)$ is linear in $X$. The proof involves expanding $g(X)$ in Hermite polynomials.