Let $X(t), 0 \leqq t$, be an $\mathscr{L} \times \mathscr{F}$-measurable process on $(\Omega, \mathscr{F}, P)$ with state space $(E, \mathscr{E})$, where $\mathscr{L}$ is the Lebesgue $\sigma$-field and $\mathscr{E}$ is countably generated. Let $\mathscr{F}(t_1, t_2), 0 \leqq t_1 < t_2 \leqq \infty$, be the $\sigma$-field generated by $\{\int^t_{t_1}f(X(s)) ds, t_1 < t < t_2, 0 \leqq f \in \mathscr{E}\}$. A new process $Z(t)$ is constructed whose values consist of conditional probabilities in the wide sense over $\mathscr{F}(t, \infty)$ given $\mathscr{F}(0, t+)$. It is shown that $Z(t)$ is a homogeneous strong-Markov process on a compact metric space, with right-continuous paths having left limits for $t > 0. Z(t)$ determines $X(t) \mathrm{wp} 1$ except for $t$ in a Lebesgue-null set. We call $Z(t)$ the prediction process of $X(t)$. Some general properties of the construction are developed, followed by two applications.

Suppose that the $k$-dimensional lattice points are displaced independently of each other with the same probability distribution. Denote by $\operatorname{Var} N(A)$ the variance of the number of displaced points contained in the set $A$. The asymptotic behaviour of $\operatorname{Var} N(A)$ is determined for large convex $A$'s.

Statistics connected with the maximum of moving averages of series of i.i.d. random variables are investigated.

Let $\mathscr{L}$ be the space of processes, progressively measurable with respect to an increasing family of $\sigma$-algebras $\{\mathscr{F}_t\}$ and having finite mean. Then $\mathscr{J}(s)f(t) = E(f(t + s) \mid \mathscr{F}_t), f \in \mathscr{L}$, defines a semigroup of linear operators on $\mathscr{L}$. Using $\mathscr{J}(s)$ and known semigroup approximation theorems, techniques are developed for proving convergence in distribution of a sequence of (possibly non-Markov) processes to a Markov process. Results are also given which are useful in proving weak convergence. In particular for a sequence of Markov process $\{X_n(t)\}$ it is shown that if the usual semigroups $(T_n(t)f(x) = E(f(X_n(t)) \mid X(0) = x))$ converge uniformly in $x$ for $f$ continuous with compact support, then the processes converge weakly.

A theorem exhibiting the duality between certain infinite systems of interacting stochastic processes and a type of branching process is proved. This duality is then used to study the ergodic properties of the infinite system. In the case of the vector model a complete understanding of the ergodic behavior is obtained.

Let $\{S_n = \sum^n_{i=1} Y_i\}$ or $\{X_t, t \geqq 0\}$ be a stochastic process with stationary independent increments, and let $T^+(\tau), T^-(\tau)$ be the times elapsed until the process has spent time $\tau$ at its maximum and minimum respectively, defined in terms of local time when necessary. Bounds in terms of moments of $Y_1$ or $X_1$ are given for $E(\min (T^+(\tau), T^-(\tau)))$. The discrete case is studied first and the result for continuous-time processes is obtained by a limiting argument. As an auxiliary it is shown that the local time at zero of a process $X_t$ minus its maximum can be approximated uniformly in probability using the number of new maxima attained by the process observed at discrete times.

Let $\{X_n, n = 1, 2, \cdots\}$ be a Markov chain with finite state space $S = \{1, 2, \cdots, d\}$, transition probability matrix $P$ and initial distribution $p$. Let $g$ be a function with $S$ as domain and define $Y_n = g(X_n)$. Define \begin{align*}Z_n^i &= \operatorname{Pr}\lbrack X_n = i \mid Y_1, Y_2, \cdots, Y_n \rbrack, \\ Z_n &= (Z_n^1, Z_n^2, \cdots, Z_n^d),\end{align*} and let $\mu_n$ denote the probability distribution of the vector $Z_n$. In this paper we prove that if $\{X_n, n = 1, 2, \cdots\}$ is ergodic and if $P$ and $g$ satisfy a certain condition then $\mu_n$ converges to a limit and this limit is independent of the initial distribution $p$.

In this paper it is shown that if $q$ is the density of a symmetric stable density, then for $c \in (0, 1) \cup (1, \infty)$, the graph of $q(x)$ intersects the graph of $cq(cx)$ at only two points. The argument proceeds by introducing a new characterization of unimodality for densities and involves a representation for symmetric stable random variables that is also useful for simulating such random variables. Finally our results are applied to prove some inequalities concerning the total variation norm of the difference of two symmetric stable densities.

In this paper, the concept of $\varphi$-mixing is extended to random fields, and a central limit theorem analogous to Theorem 20.1 of Billingsley (Convergence of Probability Measures, Wiley (1968)) is obtained for stationary, $\varphi$-mixing random fields.

This note extends to a broad class of stochastic processes with Gaussian increments the following theorem of R. M. Dudley (Ann. Probability 1 66-103): if $\{\pi_n\}$ is any sequence of partitions of [0, 1] with mesh $(\pi_n) = o(1/\log n)$ and if $L(\pi_n)^2$ is the quadratic variation of Brownian motion corresponding to $\pi_n$, then a.s.-$\lim_{n\rightarrow\infty}L(\pi_n)^2 = 1$. (Actually, Dudley proves a more general theorem). The main tool used is a bound of exponential type for the tail probabilities of quadratic functions of Gaussian random variables (Hanson and Wright, Ann. Math. Statist. 42 1079-1083).

An example is given of a gambling problem such that, on the one hand, for every initial state, there is a strategy which brings one to the goal with arbitrarily high probability and, on the other hand, for some initial state, every stationary strategy reaches the goal with probability zero.

A sum of nonnegative integer-valued random variables may be treated as a Poisson variable if the summands have sufficiently high probabilities of taking 0 value and sufficiently weak mutual dependence. This paper presents simple exact upper bounds for the error of such an approximation. An application is made to obtain a new extension for dependent events of the divergent part of the Borel-Cantelli lemma. The bounds are illustrated for the case of Markov-dependent Bernoulli trials. The method of the paper is to reduce the general problem to the special case of independent 0-1 summands and then make use of known bounds for this special case.

An entropy zero $\times$ Bernoulli process is a stationary finite state process whose shift transformation is the direct product of an entropy zero transformation and a Bernoulli shift. We show that the class of such transformations which are ergodic is closed in the $\bar{d}$-metric. The $\bar{d}$-metric measures how closely two processes can be joined to form a third stationary process.

If $k$ is a positive odd integer, it is shown that it is possible to construct a characteristic function $f(t)$ such that $f^{(k)}(0)$ exists but $f^{(k)}(t_m)$ does not exist for a sequence of numbers $\{t_m\}$ where $t_m \rightarrow 0$ as $m \rightarrow \infty$.