Let $Y_t$ be a stochastic process starting at $y$ which changes by i.i.d. dichotomous increments $X_t$ with mean 0 and variance 1. The cost of proceeding one step is one and the payoff is zero unless $n$ steps are taken and the final value $\hat{Y}$ of $Y_t$ is negative in which case the payoff is $\hat{Y}^2$. The optimal procedure consists of stopping as soon as $Y_t \geq \tilde{y}_m$ where $m$ is the number of steps left to be taken. The limit of $\tilde{y}_m$ as $m \rightarrow \infty$ is desired as a function of $p = P(X_t < 0)$. This limit $\tilde{y}$ is evaluated for $p$ rational and proved to be continuous in $p$. One can use $\tilde{y}$ to relate the solution of optimal stopping problems involving a Wiener process to those involving certain discrete-time discrete-process stopping problems. Thus $\tilde{y}$ is useful in calculating simple numerical approximations to solutions of various stopping problems.

In this paper we investigate the almost sure convergence of the Robbins-Monro process $x_{n+1} = x_n - a_n(y_n - \alpha)$ under assumptions about the conditional distribution of $y_n$ given $x_n$ which involve the existence of first moments or something closely related. The process $x_n$ can converge almost surely even when the series $\sum^\infty_{n=1} a_n\lbrack y_n - E\{y_n\mid x_n\} \rbrack$ does not do so.

Let $(X_n(t): t \in \lbrack 0, 1 \rbrack, n \geqq 1)$ be a sequence of Gaussian processes with mean zero and continuous paths on [0, 1] a.s. Let $R_n(t, s) = EX_n(t)X_n(s)$ and suppose that $(R_n: n \geqq 1)$ is uniformly convergent, on the unit square, to a covariance function $R$. It is shown in this paper that under certain conditions the normalized sequence $(Y_n(t): t \in \lbrack 0, 1 \rbrack, n \geqq 2)$ where $Y_n(t) = (2\lg n)^{-\frac{1}{2}}X_n(t)$ is, with probability one, a sequentially compact subset of $C\lbrack 0, 1 \rbrack$ and its set of limit points coincides a.s. with the unit ball in the reproducing kernel Hilbert space generated by $R$. This is Strassen's form of the iterated logarithm in its intrinsic formulation and includes a special case studied by Oodaira in a recent paper.

The equivalence between absorbing and reflecting barrier problems for random walks is shown to hold for stochastically monotone Markov processes. For Markov chains in continuous time this relation is expressed directly in terms of the $Q$-matrices of the chains. Some examples are given.

For a pair of random variables, $(X, Y)$ on the space $\mathscr{X} \times \mathscr{Y}$ and a positive constant, $\lambda$, it is an important problem of information theory to look for subsets $\mathscr{A}$ of $\mathscr{X}$ and $\mathscr{B}$ of $\mathscr{Y}$ such that the conditional probability of $Y$ being in $\mathscr{B}$ supposed $X$ is in $\mathscr{A}$ is larger than $\lambda$. In many typical situations in order to satisfy this condition, $\mathscr{B}$ must be chosen much larger than $\mathscr{A}$. We shall deal with the most frequently investigated case when $X = (X_1,\cdots, X_n), Y = (Y_1,\cdots, Y_n)$ and $(X_i, Y_i)$ are independent, identically distributed pairs of random variables with a finite range. Suppose that the distribution of $(X, Y)$ is positive for all pairs of values $(x, y)$. We show that if $\mathscr{A}$ and $\mathscr{B}$ satisfy the above condition with a constant $\lambda$ and the probability of $\mathscr{B}$ goes to 0, then the probability of $\mathscr{A}$ goes even faster to 0. Generalizations and some exact estimates of the exponents of probabilities are given. Our methods reveal an interesting connection with a so-called hypercontraction phenomenon in theoretical physics.

Consider a queueing system with infinitely many servers, a general distribution of service times and an instantaneous rate $\alpha_k$ of new arrivals, where $\alpha_k$ depends only on the number of busy servers. This is called a generalized Erlang model (GEM) since if $\alpha_k = \alpha (k < N), \alpha_k = 0 (k \geqq N)$, then Erlang's model for a telephone exchange with $N$ lines is recovered. The synchronous and asynchronous stationary distributions of the GEM are determined and several interesting properties of the process are discussed. In particular the stationary GEM is shown to be reversible.

In 1972, starting from a transient Markov process with a nice semigroup satisfying the absolute continuity hypothesis, P. A. Meyer and I built a nice dual semigroup and then obtained a Martin compactification modulo a polar set. Now, in this paper, we start from this Martin space and study the behavior of the sample paths. We prove that the Martin boundary so constructed appears in the classical form which allows one to describe the final behavior of the sample paths. We also prove that the Martin boundary we construct is an "entrance boundary" such as the Ray boundary. Finally we study a class of additive functionals which ignore the discontinuities of the process in the Martin space and which constitute a nice class of "natural" additive functionals. From all this we conclude that our Martin space is better suited for the study of the process than either the Ray space or the original space.

Bounds on absorption probabilities for Banach space-valued Brownian motion are obtained as expectations of estimates for the conditional probability given the endpoint of the path. The results are applied to the problem of computing the tail distributions of the supremum, $S$, of the two-parameter Wiener process and the supremum, $S'$, of its tied-down version. It is shown that for $\lambda \geqq 0$, $$P\{S' \geqq \lambda\} \geqq (2\lambda^2 + 1)\exp\lbrack -2\lambda^2 \rbrack$$ and $$P\{S \geqq \lambda\} \geqq 4 \int^\infty_\lambda sN(-s)ds$$ where $N(s)$ denotes the standard normal distribution. A corollary is that $P(S \geqq \lambda) \approx 4N(-\lambda)$ as $\lambda \rightarrow + \infty$.

The foundations of point process theory are surveyed. An abstract theory motivated by applications in stochastic geometry is presented. It is shown that it is sufficient to know only which sets are measurable and which are bounded in the basic space, where we use countability hypotheses rather than topological assumptions. (The sole exception is in the construction of probabilities where pseudo-topological hypotheses are needed.) It is shown that there are close connections with the random set theories of Kendall and Matheron.

We show how to build models of random collections of geometrical objects: lines, circles, line segments, etc. This basic problem in stochastic geometry is solved using the theory of point processes on abstract spaces.

This paper applies ideas from random set theory to simple point processes. We show stationarity of the hitting distributions suffices for the strict stationarity of a simple point process, but that in general all forms of stationarity differ. We compare and contrast the superposition operations of summation for random measures and union for random sets, specialized to point processes. Finally we consider completely random sets and their factors.

The current proof of the unimodality of all stable densities uses general harmonic function arguments to interpolate unimodality between the symmetric case and the totally asymmetric case. It is here shown that the proof is invalid.

Probability density estimates are generated by a kernel or weight function. Limit theorems are obtained for the maximum of the normalized deviation of the estimate from its expected value. The results are, in part, an extension to the $k$-dimensional case $(k > 1)$ of those obtained by P. Bickel and M. Rosenblatt (Ann. Statist. 1973, 1071-1095) one dimensionally.

Let $(F_n)_{n=1,2,\cdots}$ be a sequence of sigma-fields on a set $\Omega$, each $F_n$ purely atomic with respect to a measure $P$. Let $C$ denote a nested sequence of sets $C_n$, where $C_n$ is a $P$-atom of $F_n$ for each $n$. Define $S(C) = \Sigma_n(P(C_n - C_{n+1})/P(C_n))$. Then every $L^1$-bounded martingale relative to $(F_n)_{n=1,2,\cdots}$ and $P$ is uniformly integrable if and only if $S$ is finite-valued, and every such martingale is dominated if and only if $S$ is uniformly bounded.

A simple geometric proof and some applications are given to results of C. Borell providing necessary and sufficient conditions that a density in $R^n$ generates a measure satisfying a convexity property of the type $$P(\theta A_0 + (1 - \theta)A_1) \geqq \{\theta\lbrack P(A_0)\rbrack^s + (1 - \theta)\lbrack P(A_1) \rbrack^s\}^{1/s}.$$

We discuss the convergence of the expected times until the partial sums of a sequence of independent, identically distributed random variables with zero means and unit variances first rise a height $h$ above their previous minimum as $h \rightarrow \infty$. We also consider the convergence as $r \rightarrow \infty$ of the expected times until the range of these partial sums exceeds a value $r$. Applications of these results to a quality control procedure and to queueing theory are mentioned.

The class of IFRA distributions is closed under convolution.

Among the class of totally ergodic stationary discrete stochastic processes, the Bernoulli processes are characterized by $\bar{d}$-convergence of their canonical $k$-step Markov approximations. Here this property is shown to be no longer invariant under isomorphism if we leave the totally ergodic class. On the other hand, the isomorphism class of all Markov chains is shown to be not closed under $\bar{d}$-convergence.