Results are obtained on synchronizing block codes for discrete stationary totally ergodic $\bar{d}$-continuous noisy channels (which may have infinite memory and anticipation) and used to prove sliding-block joint source and channel coding theorems. The coding theorems are used to demonstrate the existence of invulnerable sources--ergodic sources which can be input directly to the channel without encoding and decoded at the receiver with zero error--at all entropy rates below channel capacity. Combining the invulnerable source theorem with the isomorphism theorem of ergodic theory shows that, if the source is a $B$-process with entropy below capacity, then infinite length codes with zero error exist, proving that the zero-error capacity equals the usual channel capacity.

A natural generalization of the optional sampling theorem for martingales is given. For discrete valued stopping times the result holds for directed sets; for more general stopping times the result holds for lattices satisfying a type of separability condition. The discrete case improves a lemma of Chow. The general case depends upon a lemma showing that all martingales with respect to $\sigma$-algebras satisfying a "right continuity" condition have a modification which has a regularity property that is similar to, but weaker than, right continuity. A result of Wong and Zakai is obtained as a corollary.

Let $Y_1, Y_2,\cdots$ be independent Markov processes. Solutions of equations of the form $Z_i(t) = Y_i(\int^t_0\beta_i(Z(s))ds)$, where $\beta_i(z) \geqslant 0$, are considered. In particular it is shown that, under certain conditions, the solution of this "random time change problem" is equivalent to the solution of a corresponding martingale problem. These results give representations of a large class of diffusion processes as solutions of $X(t) = X(0) + \sum^N_{i=1}\alpha_iW_i(\int^t_0\beta_i(X(s))ds)$ where $\alpha_i \in \mathbb{R}^d$ and the $W_i$ are independent Brownian motions. A converse to a theorem of Knight on multiple time changes of continuous martingales is given, as well as a proof (along the lines of Holley and Stroock) of Liggett's existence and uniqueness theorems for infinite particle systems.

We start with a standard Markov process $X$ and a continuous additive process $A$ of $X$ with fine support $\Phi$. We form the time changed process $X_\tau$, and we compute its weak infinitesimal generator in terms of the weak infinitesimal generator of the process $X$ and of the Levy system of $(X_\tau, \tau)$. We give some examples.

Let $M_t$ be the number of males and $F_t$ the number of females present at time $t$ in a population where births take place at rates which at time $t$ are $mR(M_t, F_t)$ and $fR(M_t, F_t)$ for males and females, respectively. Assume that $R$ has the form $R(M, F) = (M + F)h(M/(M + F))$ with $h$ sufficiently smooth at $m/(m + f)$. A Malthusian parameter $\lambda$ and a random variable $W$ such that $e^{-\lambda t}M_t\rightarrow mW, e^{-\lambda}F_t\rightarrow fW$ a.s. are exhibited, the rate of convergence is found in form of a central limit theorem and a law of the iterated logarithm and an asymptotic expansion of the reproductive value function $\tilde{V}(M, F) = E(W\mid M_0 = M, F_0 = F)$ is given. Also some discussion of an associated set of deterministic differential equations is offered and the stochastic model compared to the solutions.

Let $X_1, X_2,\cdots, X_k, X_{k+1},\cdots, X_n$ be exchangeable random variables taking values in the set $S$. The variation distance between the distribution of $X_1, X_2,\cdots, X_k$ and the closest mixture of independent, identically distributed random variables is shown to be at most $2 ck/n$, where $c$ is the cardinality of $S$. If $c$ is infinite, the bound $k(k - 1)/n$ is obtained. These results imply the most general known forms of de Finetti's theorem. Examples are given to show that the rates $k/n$ and $k(k - 1)/n$ cannot be improved. The main tool is a bound on the variation distance between sampling with and without replacement. For instance, suppose an urn contains $n$ balls, each marked with some element of the set $S$, whose cardinality $c$ is finite. Now $k$ draws are made at random from this urn, either with or without replacement. This generates two probability distributions on the set of $k$-tuples, and the variation distance between them is at most $2 ck/n$.

Strong laws of large numbers, obtained for positive, independent random variables, are utilized to prove iterated logarithm laws (with a nonrandom normalizing sequence) for a class of martingales. A law of the iterated logarithm is also established for certain random quadratic forms.

A level crossing predictor is a predictor process $Y(t)$, possibly multivariate, which can be used to predict whether a specified process $X(t)$ will cross a predetermined level or not. A natural criterion on how good a predictor is, can be the probability that a crossing is detected a sufficient time ahead, and the number of times the predictor makes a false alarm. If $X$ is Gaussian and the process $Y$ is designed to detect only level crossings, one is led to consider a multivariate predictor process $Y(t)$ such that a level crossing is predicted for $X(t)$ if $Y(t)$ enters some nonlinear region in $R^p$. In the present paper we develop the probabilistic methods for evaluation of such an alarm system. The basic tool is a model for the behavior of $X(t)$ near the points where $Y(t)$ enters the alarm region. This model includes the joint distribution of location and direction of $Y(t)$ at the crossing points.

A class of multivariate distributions which is an extension of the class of univariate distributions with increasing failure rate averages is introduced. Properties of this class are studied and examples of distributions which are members of this class are given.

Let $(X_1, X_2)$ be independent $N(0, 1)$ variables and let $P(v_1, v_2) = P\lbrack(X_1, X_2) \in C + (v_1, v_2)\rbrack$, where $C$ is the square $\{|x_1| \leqslant a,|x_2| \leqslant a\}$. By demonstrating that $P\lbrack|X_i - v_i|\leqslant a\rbrack$ is $\log$ concave in $v^2_i$, the extrema of $P(v_1, v_2)$ on all circles $\{v^2_1 + v^2_2 = b^2\}$ are determined. The results are extended to determine the extrema of the probability of a cube in $R^n$. The proof is based on a log concavity-preserving property of the Laplace transforms.

An ordering on discrete bivariate distributions formalizing the notion of concordance is defined and shown to be equivalent to stochastic ordering of distribution functions with identical marginals. Furthermore, for this ordering, $\int\varphi dH$ is shown to be $H$-monotone for all superadditive functions $\varphi$, generalizing earlier results of Hoeffding, Frechet, Lehmann and others. The usual correlation coefficient, Kendall's $\tau$ and Spearman's $\rho$ are shown to be monotone functions of $H$. That $\int\varphi dH$ is $H$-monotone holds for distributions on $\mathbb{R}^n$ with fixed $(n - 1)$-dimensional marginals for any $\varphi$ with nonnegative finite differences of order $n$. Some related results are obtained. Stochastic ordering is preserved under certain transformations, e.g., convolutions. A distribution on $\mathbb{R}^\infty$ is constructed, making $\max(X_1,\cdots, X_n)$ stochastically largest for all $n$ when $X_i$ have given one-dimensional distributions, generalizing a result of Robbins. Finally an ordering for doubly stochastic matrices is proposed.

The problem of testing uniformity on [0, 1] against a clustering alternative, is considered. Naus has shown that the generalized likelihood ratio test yields the scan statistic $N(d)$. The asymptotic distribution of $N(d)$ under the null hypothesis of uniformity is considered herein, and related to the version of the scan statistic defined for points from a Poisson process. An application of the above yields distributional results for the supremum of a stationary Gaussian process with a correlation function that is tent-like in shape, until it flattens out at a constant negative value.

An alarm system is optimal if it detects catastrophes with a certain high probability and simultaneously gives a minimum number of false alarms. In a general context an optimal alarm system is derived and then the method is applied to Gaussian processes.

It is shown that a wide-sense stationary process is strictly stationary (i) only if it is rotation invariant, (ii) if it is rotation invariant and has a (two dimensional) random spectral measure with independent increments.

Let $\tau_x$ be the first passage time of $x$ for a diffusion or birth-death process. If one starts in a reflecting state, say 0, then the distribution $P_0(\tau_x \leqslant \cdot)$ is strongly unimodal. Here we show for an arbitrary state 0 the distribution $P_0(\tau_x \leqslant \cdot)$ is unimodal. Further we give a discrete analogue for the random walk.