The Wald Memorial Lectures delivered in Seattle in August 1977, and summarised here, ranged over a variety of applications in both pure and applied probability of the idea of exchangeability, and particularly of de Finetti's theorem. Particular emphasis was placed on two contrasting themes, some recent work of Aldous on the subsequence principle, and consequences of de Finetti's theorem for certain problems in population genetics.

A new proof is given that a contact process on $Z^d$ has a nontrivial stationary measure if the birth rate is sufficiently large. The proof is elementary and avoids the use of percolation processes, which played a key role in earlier proofs. It yields upper bounds for the critical birth rate which are significantly better than those available earlier. In one dimension, these bounds are no more than twice the actual value, and they are no more than four times the actual critical value in any dimension. A lower bound for the particle density of the largest stationary measure is also obtained.

If $d$ is a fixed positive integer, let $\Lambda_N$ be a finite subset of $Z^d$, the lattice points of $\mathbb{R}^d$, with $\Lambda_N \uparrow Z^d$ and satisfying certain regularity properties. Let $(X_{N, Z})_{Z\in\Lambda_N}$ be a collection of random variables which satisfy a mixing condition and whose partial sums $X_N = \sum_{Z\in\Lambda_N} X_{N, Z}$ have uniformly bounded variances. Limit theorems, including a central limit theorem, are obtained for the sequence $X_N$. The results are applied to Gibbs random fields known to satisfy a sufficiently strong mixing condition.

A class of $\mathscr{D}$ of random measures, generalizing the class of completely random measures, is developed and shown to contain the class of Poisson cluster point processes. An integral representation is obtained for $\mathscr{D}$, generalizing the Levy-Ito representation for processes with independent increments. A subclass $\mathscr{D}_n \subset \mathscr{D}$ is defined such that for $X \in \mathscr{D}_n$, the distribution of the random vector $X(A_1), \cdots, X(A_m), m > n, A_1, \cdots, A_m$ disjoint, is determined by the distributions of all subvectors $X(A_{i_1}), \cdots, X(A_{i_k}), 1 \leqq k \leqq n$. The class $\mathscr{D}_1$ coincides with the class of completely random measures.

For $n \geqq 2$ let $X_{nij}, i, j = 1, \cdots, n$, be a square array of independent random variables with finite variances and let $\pi_n = (\pi_n(1), \cdots, \pi_n(n))$ be a random permutation of $(1, \cdots, n)$ independent of the $X_{nij}$'s. By using Stein's method, a bound is obtained for the $L_p$ norm $(1 \leqq p \leqq \infty)$ with respect to the Lebesgue measure of the difference between the distribution function of $(W_n - EW_n)/(\operatorname{Var} W_n)^{\frac{1}{2}}$ and the standard normal distribution function where $W_n = \sum^n_{i=1} X_{ni\pi_n(i)}$. This result generalizes and improves a number of known results. In particular, it provides bounds for Motoo's combinatorial central limit theorem as well as the central limit theorem.

Some fundamental properties of the empirical distribution function are derived in the case of independent but not necessarily identically distributed random variables. The distribution functions of these random variables need not be continuous.

We prove that an amart indexed by a directed set decomposes into a martingale and an amart which converges to zero in $L_1$ norm. The main theorem asserts that the underlying family of $\sigma$-algebras satisfies the Vitali condition if and only if every $L_1$ bounded amart essentially converges.

The subclasses of class $L$ probability distributions recently studied by K. Urbanik are characterized by requiring that certain functions be convex and have derivatives of some fixed order. The extreme points of certain compact convex sets of probability measures are determined, and this information is then used to obtain a representation of the characteristic functions of the probability distributions in those classes, in the same manner as Urbanik has proceeded for the class $L$.

An elementary probabilistic argument is given which establishes a "weak invariance principle" which in turn implies the sufficiency of the classical assumptions associated with the weak convergence of normed sums to stable laws. The argument, which uses quantile functions (the inverses of distribution functions), exploits the fact that two random variables $X = F^{-1}(U)$ and $Y = G^{-1}(U)$ are, in a useful sense, close together when $F$ and $G$ are, in a certain sense, close together. Here $U$ denotes a uniform variable on (0, 1). By-products of the research are two alternative characterizations for a random variable being in the domain of partial attraction to a normal law and some results concerning the study of domains of partial attraction.

The co-thinness of sets is studied by means of $h$-processes. Co-thin sets are characterized by a balayage property and are semipolar sets. The method avoids the use of time reversal.

A new class of functions, the BP (Borel-programmable) functions, is defined. It is strictly larger than the class of Borel functions, but has some similar properties, including closure under composition. All BP functions are absolutely measurable. The class of BP sets (those with BP indicators) is a Borel field and is closed under operation A. The relation of BP sets to the R-sets of Kolmogorov is not treated.

Renyi introduced and developed the ideas of limit theorems which are mixing or stable. These concepts are a strengthening of the idea of weak convergence of random variables. In this expository note we point out some equivalent definitions of mixing and stability and discuss the use of these concepts in several contexts. Further, we show how a recent central limit theorem for martingales can be obtained directly using stability. Though the results are not new, the proofs seem substantially simpler than those previously given.

Every measurable real-valued function $f$ on the space of Wiener process paths with $E(|f|^p) < \infty$ (where $0 < p < 1$) can be represented as a stochastic integral $f = \int \varphi dX$, where $E(\int \varphi^2(t)dt)^{p/2} < \infty$. A similar result holds for $1 < p < \infty$ if and only if $E(f) = 0$.

A sufficient condition for the tightness of a sequence of stochastic processes is given in terms of their behavior after stopping times. As an application, the conditions for McLeish's invariance principle for martingales are weakened.

Let $\bar{F}_n(x)$ denote the distribution of the normalized partial sum of independent, identically distributed random variables with finite second moment, and write $\Delta_n(x) = |\bar{F}_n(x) - \Phi(x)|$, where $\Phi(x)$ is the standard normal distribution. In this paper, the necessary and sufficient conditions for the validity of $\|(1 + |x|)^{2 - 1/p}\Delta_n(x)\|_p = O(n^{-\delta/2})$ and of $\sum n^{-1 + \delta/2}\|(1 + |x|)^{2 - 1/p}\Delta_n(x)\|_p < \infty, 0 < \delta < 1, 1 \leqq p \leqq \infty$, are given. Furthermore, in the case where the underlying random variables $\{X_k\}$ are independent but not necessarily identically distributed, it is shown that $E|X_k|^{2 + \delta} < \infty$ implies $\|(1 + |x|)^{2 + \delta - 1/p}\Delta_n(x)\|_p \leqq Cs_n^{-(2 + \delta)} \sum^n_{k = 1} E|X_k|^{2 + \delta}, 0 < \delta < 1, 1 \leqq p \leqq \infty$.

Let $X = \{X_t, t > 0\}$ be a right continuous strong Markov process with state space $E$; let $f$ be a continuous real valued function on $E \times E$; and let $M$ be the time at which the process $\{f(X_{t-}, X_t)\}$ achieves its (last) ultimate minimum. Then conditional on $X_M$ and the value of this minimum, the process $\{X_{M + t}\}$ is Markov and (conditionally) independent of events before $M$.

Let $\Gamma_n$ denote the empirical df of a sample from the uniform (0, 1) df $I$. Let $\xi_{nk}$ denote the $k$th smallest observation. Let $\lambda_n > 1$. Let $A_n$ denote the event that $\Gamma_n$ intersects the line $\lambda_n I$ on [0, 1] and let $B_n$ denote the event that $\Gamma_n$ intersects the line $I/\lambda_n$ on $\lbrack\xi_{n1}, 1\rbrack$. Conditions on $\lambda_n$ are given that determine whether $P(A_n \mathrm{i.o.})$ and $P(B_n \mathrm{i.o.})$ equal 0 or 1. Results for $A_n$ (for $B_n$) are related to upper class sequences for $1/(n\xi_{n1})$ (for $n\xi_{n2})$. Upper class sequences for $n\xi_{nk}$, with $k > 1$, are characterized. In the case of nonidentically distributed random variables, we present the result analogous to $P(A_n \mathrm{i.o.}) = 0$.