Phase transition is studied on the infinite tree $T_N$ in which every point has exactly $N + 1$ neighbors. For every assignment of conditional probabilities which are invariant under graph isomorphism there is a Markov chain with these conditional probabilities and the main results ascertain for which ones of these chains there are other Markov random fields with the same conditional probabilities.

We associate with a strong Markov process $(X_t)$ and a Borel set $B$ an "exit system." This system provides the structure of the excursions from $B$ of the process $(X_t)$ and gives a new approach to the recent results of Getoor and Sharpe on last exit decompositions and last exit distributions.

In prediction (Wiener-, Kalman-) of a random normal process $\{X(t), t \in R\}$ it is normally required that the time $t_0$ from which prediction is made does not depend on the values of the process. If prediction is made only from time points at which the process takes a certain value $u,$ given a priori, ("prediction under panic"), the Wiener-prediction is not necessarily optimal; optimal should then mean best in the long run, for each single realization. The main theorem in this paper shows that when predicting only from upcrossing zeros $t_\nu$, the Wiener-prediction gives optimal prediction of $X(t_\nu + t)$ as $t_\nu$ runs through the set of zero upcrossings, if and only if the derivative $X'(t_\nu)$ at the crossing points is observed. The paper also gives the conditional distribution from which the optimal predictor can be computed.

Some ideas from the theory of weak convergence of probability measures on function spaces are modified and extended to show that the queue-length of the GI/G/s system converges in distribution as time passes, for the case of atomless interarrival and service distributions. The key to this result is the concept of the uniform $\sigma$-additivity of certain sets of renewal measures on a space endowed with incompatible topology and $\sigma$-field.

A Markov Process is called semi-stable if it is invariant under certain dilation of time and space. The particular case of semi-stable continuous branching Markov Processes is studied.

We obtain representation formulas for the Radon-Nikodym derivatives of measures absolutely continuous with respect to measures induced by processes with stationary independent increments. The proofs of these formulas, which have applications in signal detection and estimation problems, call heavily upon recent results in martingale theory, especially a general formula of Doleans-Dade for the logarithm of a strictly positive martingale in terms of a function measuring its jumps.

This paper is concerned with the existence of bounded solutions to an operator inequality which is a non-linear version of a discrete time diffusion equation. Here, the solvability of the inequality will be closely related to the transience of a corresponding random walk. In particular, the inequality will generally be solvable in three or more dimensions, but not in one or two dimensions if appropriate moment conditions hold.

A new nonblock source coding (data compression) technique is introduced and a source coding theorem is proved using recently developed techniques from ergodic theory. The existence theorem is valid for all stationary aperiodic sources (e.g., ergodic sources) with finite alphabets and all ergodic sources with separable alphabets and is proved without Shannon-style random coding arguments. The coding technique and the optimal performance bounds are compared and contrasted with Shannon block coding techniques.

A mass $m(x) \geqq 0$ is assigned to each point $x$ of a partially ordered countable set $X$. It is further assumed that $M(x) = \sum_{y\leqq x} m(y) < \infty$ for each $x \in X. M$ is called a distribution function. For certain sets $X$, it is shown that $M$ determines $m$. For others, $M$ need not determine $m$ uniquely. A theory is presented for $\sigma$-lower finite spaces (sets), which are defined in the paper. Such spaces are locally finite. That is, each interval $\lbrack x, y \rbrack = \{z \in X: x \leqq z \leqq y\}$ has a finite number of points. Mobius functions, which have been defined for locally finite spaces, are used throughout. Distribution functions on a particular $\sigma$-lower finite space arise naturally from boundary crossing problems analyzed by Doob and Anderson. The theory is applied to this example and to another.

Let $X_1, X_2, \cdots$ be i.i.d. random vectors in $R_k$. Let $P_n$ denote the probability measure induced by the normalized sum and let $Q_n$ denote the multivariate Edgeworth signed measure with terms through $n^{-\frac{1}{2}}$. If $C$ is a member of a class of convex bodies whose boundaries are sufficiently smooth and possess positive Gaussian curvatures, and $X_1$ has fourth moments, it is shown that $P_n(C) - Q_n(C) = 0(n^{-k/(k + 1)})$ where the bound is uniform. If, moreover, $X_1$ has a nonlattice distribution, the difference is $o(n^{-k/(k + 1)})$.

For a sequence $\{X_n: 1 \leqq n < \infty\}$ of independent, identically distributed random variables with moment-generating functions, a 1952 theorem of Chernoff asserts that $n^{-1} \log P(S_n \geqq \lambda n) \rightarrow \log \rho$, where $S_n$ is the $n$th partial sum of the $X_k's \lambda > 0$, and $\rho$ is a constant depending on $\lambda$ and the distribution of $X_1$. A 1969 theorem of Sievers, as strengthened by Plachky in 1971, established the convergence of $n^{-1}\log P(W_n \geqq z_n)$ to a constant, where the $W_n$'s have moment-generating functions and belong to a class of random variables more general than partial sums, and the $z_n$'s are numbers such that $n^{-1}z_n \rightarrow \lambda > 0$. In a format related to that of Sievers, Behadur in 1971 analyzed the behavior of $n^{-1}\log P(W_n \geqq z_n)$ in situations when it may not converge to a constant. The goal of the present article is to extend the theorems of Chernoff, Sievers, and Bahadur in the direction of obtaining convergence rates (to 0) of the large deviation probabilities $P(W_n \geqq z_n)$ where the $z_n$'s are numbers such that $n^{-\frac{1}{2}} z_n \rightarrow \infty$. The method of proof is based on the proof of Chernoff's theorem given, in passing, in a 1960 paper of Bahadur and Ranga Rao.

Let $X_1, \cdots, X_n$ be an arbitrary sequence of dependent Bernoulli random variables with $P(X_i = 1) = 1 - P(X_i = 0) = p_i.$ This paper establishes a general method of obtaining and bounding the error in approximating the distribution of $\sum^n_{i=1} X_i$ by the Poisson distribution. A few approximation theorems are proved under the mixing condition of Ibragimov (1959), (1962). One of them yields, as a special case and with some improvement, an approximation theorem of Le Cam (1960) for the Poisson binomial distribution. The possibility of an asymptotic expansion is also discussed and a refinement in the independent case obtained. The method is similar to that of Charles Stein (1970) in his paper on the normal approximation for dependent random variables.

The Wold representation of wide-sense stationary processes with continuous parameter is obtained from a general Cramer representation of second-order processes, continuous in quadratic mean.

Expressions for first and last passage probabilities, conditioned on both an initial and a subsequent value, for the Gaussian process with triangular covariance and mean zero, are derived. We use these to bound the passage probabilities of arbitrary functions, to derive formulas for the expectation of passage and excursion times, to prove the uniqueness of the maximum of the process in the interval $\lbrack 0, 1 \rbrack$, and to find a formula for the joint probability density function of the maximum and its instant of occurrence.

Sufficient conditions are found for the row sums of a double array of martingale differences to converge in law to a mixture of infinitely divisible distributions.

Any infinitely divisible distribution on $R^n$ with infinite absolutely continuous Levy measure and no Gaussian component has a density which is positive a.e. over its support.

A certain sequence of statistics based on sample spacings is shown to constitute a martingale. This fact is applied to prove an inequality relating to a process of picking points randomly but not independently.