An $n$-code is a mapping $c_n$ from the set $A^n$ of sequences of length $n$ drawn from a finite set $A$ into the set of finite length binary sequences $B^\ast$. A decoder with distortion $D$ is a map from $B^\ast$ back into $A^n$ that sends $c_n(a^n_1)$ into a sequence that agrees with $a^n_1$ in all but at most $Dn$ places. We describe a sequence of codes and associated decoders of distortion $D$ such that, for almost every sequence from an ergodic process, the number of bits per $A$-symbol converges almost surely to $R(D)$, the optimal compression attainable for the process. The codes are universal in that the statistics of the process need not be known in advance. Expected value results of this type were first obtained by Davisson and, independently, Fittinghof; almost sure results for the invertible case $(D = 0)$ are implicitly contained in the Ziv-Lempel algorithm. Our results also apply, virtually without change in proof, to random fields.

We study the closed support of the measure-valued diffusions of Watanabe and Dawson. When the spatial motion is Brownian, sufficient conditions involving capacity are given for a fixed set to be hit by the $k$-multiple points of the support process. The conditions are close to the necessary conditions found by Dawson, Iscoe and Perkins and lead to necessary and sufficient conditions for the existence of $k$-multiple points. When the spatial motion is a symmetric stable process of index $\alpha < 2$, the closed support is shown to be $\mathbb{R}^d$ or $\varnothing$.

Let $X_t$ be a transient stable process on $\Re^d$ and let $T_B = \inf\{t > 0: X_t \in B\}$ be the hitting time of $B$. Set $E_B(t) = \int P_x(T_B \leq t) dx$. Asymptotic expansions, as $t \rightarrow \infty$, to order 3 are obtained for all stable processes on $\Re$ that are not completely asymmetric and for all strictly stable processes on $\Re^d, d \geq 2$, whose transition density at time 1 is not zero at the origin. For those processes that are strongly transient, nontrivial $O$ estimates of the error are also obtained. Expansions to order 2 together with $O$ estimates of the error are given for the completely asymmetric processes on $\Re$, the strictly stable processes on $\Re^d$ whose transition density vanishes at 0 at time 1 and for linear Brownian motion with nonzero mean. Asymptotic expansions to order 3 together with $O$ estimates of the error are given for stable processes with drift on $\Re^d$ having exponent $\alpha < 1$. Expansions to order 3 are also given for stable processes with drift on $\Re^d$ having exponent $\alpha > 1$ when the associated drift free process is isotropic, and expansions to order 2 with $O$ estimate of the error are obtained for the other stable processes with drift on $\Re^d$ having exponent $\alpha > 1$.

The invariant density of diffusion processes which are small random perturbations of dynamical systems can be expanded in W.K.B. type, as the random effect disappears, in the set in which the Freidlin-Wentzell quasipotential $V(\cdot)$ is of $C^\infty$-class and each coefficient which appears in the expansion is of $C^\infty$-class.

We consider standard (Bernoulli) site percolation on $\mathbb{Z}^d$ with probability $p$ for each site to be occupied. $C$ denotes the occupied cluster of the origin and $|C|$ its cardinality. We show that for $p >$ (critical probability of the halfspace $\mathbb{Z}^{d - 1} \times \mathbb{Z}_+)$ one has $P_p\{|C| = n\} \leq \exp\{-C_1(p)n^{(d - 1)/d}\}$ for some constant $C_1(p) > 0$. This improves a recent result of Chayes, Chayes and Newman. The proof is based on a Peierls argument which shows exponential decay of the distribution of the size of an "exterior boundary" of $C$.

Stationary measures for an interactive exclusion process on $\mathbb{Z}$ are considered. The process is such that the jump rate of each particle to the empty neighboring site is $\alpha > 0$ (resp., $\beta > 0$) when another neighboring site is occupied (resp., unoccupied) by a particle, and that $\alpha \neq \beta$. According as $\alpha < \beta$ or $\alpha > \beta$ the process becomes nearest-neighbor attractive or repulsive, respectively. The method of relative entropy is used to determine the family $\mathscr{M}_{\beta/\alpha}$ of stationary measures. The member of $\mathscr{M}_\gamma$ is simply described as the probability measure having the regular clustering property which is a generalization of the exchangeable property of measures. It is shown that extremal points of $\mathscr{M}_\gamma$ are renewal measures. Thus the structure of stationary measures for the process is completely determined.

The branching annihilating process (BAP) is a special case of the branching annihilating walk of Bramson and Gray. In the BAP each particle places offspring on neighbouring sites at unit rate, but when two particles occupy the same site they annihilate each other. We show that the product measure with density 1/2 is the limit starting from any $A \neq \varnothing$. Also considered will be the DBAP (double BAP) in one dimension. In this model a particle always places offspring on both neighbouring sites. The limiting measure here depends critically on whether the initial number of particles is odd or even.

We consider a system of $N (\rightarrow \infty)$ interacting one-dimensional diffusions, in which each diffusion is assigned a random charge $(\pm 1)$, and study the behavior of the net charge distribution through space and time. The diffusion equations are a slight variation of those considered in the initial studies of "the propagation of chaos," but the interaction involves the signs of the diffusions and triplet rather than pairwise interactions. This has the effect of leading to a non-Gaussian fluctuation theory, which turns out to be close to the $P(:\Phi^4:)$ models of Euclidean quantum field theory. The main tools of the proofs involve the Stroock-Varadhan martingale theory and a general theory of $U$-statistics.

In a previous paper we proved that a necessary and sufficient condition for all martingales of a given filtration $(\mathscr{F}_t)$ to be continuous is that, for every stopping time $T$ and every $\mathscr{F}_T$-measurable random variable $X$, there exists a continuous local martingale $M$ with $M_T = X$ a.s. The aim of this paper is to study the following question: Can we choose $M \geq 0$ whenever $X \geq 0$? We also give a negative answer to Conjecture 7.1 of Harrison and Pliska.

Some new maximal-type probability inequalities are developed for discrete-time multidimensionally indexed submartingales. In particular, the basic idea of Chow is abstracted and extended. This leads to a result which yields extended Kolmogorov inequalities and strong laws, extended Hajek-Renyi type inequalities competitive with Smythe and an extended Doob inequality which is counter-intuitive to a counterexample of Cairoli.

We study direct convergence of the products $P(\theta_0) \cdots P(\theta_n)$ of random stochastic matrices. These products can be interpreted as the transition probabilities of nonhomogeneous Markov chains selected at random by a stationary "environmental" sequence $\{\theta_n\}$, in other words, a Markov chain in a random environment. Rather than make assumptions analogous to irreducibility and aperiodicity for homogeneous Markov chains, we introduce equivalence relations that allow convergence results on the equivalence classes. The classical decomposition into a cycle of periodic sets is not possible in general, so the "periodicity" in the title is meant only to be suggestive. We also examine the frequency of times of positive probability of return to a state or set.

From the symmetries contained in the collection of excessive functions of a transient Hunt process $X(t)$ on a state space $E$, we construct a quotient space $F$, a function $\Phi: E \rightarrow F$ and a time change $\tau(t)$ of $X(t)$ so that $\Phi(X(\tau_t))$ is a strong Markov process.

We establish strong and weak laws for Bahadur-Kiefer-type processes of the form $e_n + i_n$, where $i_n$ denotes the inverse of $e_n$. In particular, we provide a proof for the strong version of Theorem 1A of Kiefer (1970), together with similar results for renewal and partial sum processes.

A particle in $\mathbf{R}^d$ moves in discrete time. The size of the $n$th step is of order $1/n$ and when the particle is at a position $\mathbf{v}$ the expectation of the next step is in the direction $\mathbf{F}(\mathbf{v})$ for some fixed vector function $\mathbf{F}$ of class $C^2$. It is well known that the only possible points $\mathbf{p}$ where $\mathbf{v}(n)$ may converge are those satisfying $\mathbf{F}(\mathbf{p}) = \mathbf{0}$. This paper proves that convergence to some of these points is in fact impossible as long as the "noise"--the difference between each step and its expectation--is sufficiently omnidirectional. The points where convergence is impossible are the unstable critical points for the autonomous flow $(d/dt)\mathbf{v}(t) = \mathbf{{F}({v}}(t))$. This generalizes several known results that say convergence is impossible at a repelling node of the flow.

Consider a family of (dependent) Gaussian random variables and count the number of them that exceed some given levels. An explicit upper bound is given for the total variation distance between the distribution of this number of exceedances and a Poisson distribution having the same mean. The bound involves only means and covariances of the indicators that the variables exceed the levels. The general result is illustrated by some examples from the extreme value theory of Gaussian sequences. The bound is derived as a special case of a result obtained by the Stein-Chen method for sums of dependent Bernoulli random variables. This general result requires the existence of a certain coupling, which in the Gaussian case follows by a correlation inequality.

We obtain a probability inequality for the sup-norm distance between multivariate empirical processes and suitable constructed Poisson processes.

A unified approach to approximations of probabilities for sums of $n$ independent random vectors in $\mathbb{R}^d$ is presented based on the Edgeworth expansion of exponentially shifted vectors together with explicit bounds on the errors. Weak conditions are given under which the error bounds may be written as simple order terms in $n$. These results are used in particular to examine approximations to conditional probabilities giving a general method of approximation for these. A number of important special cases are discussed and examined numerically.

We develop several applications to almost sure limit theorems for sums of independent vector valued random variables of an isoperimetric inequality due to Talagrand. A general treatment of the classical laws of large numbers of Kolmogorov and Prokorov and laws of the iterated logarithm of Kolmogorov and Hartman and Wintner is described. New results as well as simpler new proofs of known ones illustrate the usefulness of isoperimetric methods in this context. We show further how this approach can be used in the study of limit theorems for trimmed sums of independent and identically distributed random variables.

Let $\{X_n\}$ be a sequence of independent Banach space valued random elements with partial sums $S_n = X_1 + \cdots + X_n$. Then let $T$ be any possibly randomized stopping time based on $\{X_n\}$. Fix any $\alpha > 0$ and let $\Phi(\cdot)$ be any nondecreasing continuous function on $\lbrack 0, \infty)$ with $\Phi(0) = 0$ such that $\Phi(cx) \leq c^\alpha\Phi(x)$ for all $x \geq 0, c \geq 2$. Put $S^\ast_n = \max_{1 \leq k \leq n}\|S_k\|$ and $a^\ast_n = E\Phi(S^\ast_n)$. It is proved that there exists a universal constant $c^\ast_\alpha < \infty$ depending only on $\alpha$ [and otherwise independent of $(B, \|\cdot\|), \{X_n\}, T \text{and} \Phi(\cdot)$] such that $Ea^\ast_T \leq c^\ast_\alpha E\Phi(S^\ast_T)$. As a consequence, $E\Phi(S^\ast_{T_c}) = \infty$ whenever $P(T_c < \infty) = 1$ and $c \geq c^\ast_\alpha$, where \begin{equation*}T_c = \begin{cases} \text{first} n: c^\Phi(S^\ast_n) < a^\ast_n,\\ \infty, \text{if no such} n \text{exists}.\end{cases}\end{equation*} In fact, $Ea^\ast_{T_c} = \infty$, too. An upper bound for $c^\ast_\alpha$ is constructed.

Local limit theorems are derived for sums of finite range $\mathbb{Z}$-valued potential functions of an iid random field. The resulting approximations turn out to be mixtures of standard normal densities for lattice distributions supported by residue classes of integers. The mixing weights are equal to the probability that the sum of potential functions lies in such a residue class and are nonasymptotic and computable. For finite range potential functions of a stationary Gibbsian random field with bounded and finite range interactions, conditions are given under which the global central limit theorem implies the classical local limit theorem.

Let $\{X_n, n \geq 1\}$ be a sequence of random variables taking values in $S = \{1,2, \ldots, m\}$ with distribution $p(x_1, \ldots, x_n), (p_{i1}, p_{i2}, \ldots, p_{im}), i = 1,2, \ldots$, a sequence of probability distributions on $S$, and $\varphi_n = (1/n)\log p(X_1, \ldots, X_n) - (1/n)\sum^n_{i = 1}\log p_{iX_i}$ the entropy density deviation, relative to the distribution $\prod^n_{i = 1}p_{ix_i}, \text{of} \{X_i, 1 \leq i \leq n\}$. In this paper the relation between the limit property of $\varphi_n$ and the frequency of given values in $\{X_n\}$ is studied.

Some versions of the Fisher information matrix and the Cramer-Rao inequality are considered. We study properties of the Fisher matrix such as continuity and convexity and use the Cramer-Rao functional as a variational tool to prove convergence to Gaussian laws. These concepts are generalized to non-Gaussian limiting laws.

It is proved that the bootstrapped central limit theorem for empirical processes indexed by a class of functions $\mathscr{F}$ and based on a probability measure $P$ holds a.s. if and only if $\mathscr{F} \in \mathrm{CLT}(P)$ and $\int F^2 dP < \infty$, where $F = \sup_{f \in \mathscr{F}}|f|$, and it holds in probability if and only if $\mathscr{F} \in \mathrm{CLT}(P)$. Thus, for a large class of statistics, no local uniformity of the CLT (about $P$) is needed for the bootstrap to work. Consistency of the bootstrap (the bootstrapped law of large numbers) is also characterized. (These results are proved under certain weak measurability assumptions on $\mathscr{F}$.)

Let $X$ be a random vector on $\mathbb{R}^k$ whose distribution $\mu$ belongs to the domain of normal attraction of some operator stable law $\nu$. For a given $\nu$ it has been shown elsewhere that for certain ranges of $\alpha$ depending on $\nu$, either $E|\langle X, \theta\rangle|^\alpha$ is finite for every $\theta \neq 0$ or is infinite for every $\theta \neq 0$. In this paper we show that the set of $\alpha$ for which $E|\langle X, \theta\rangle|^\alpha$ exists depends, in general, on both $\theta$ and $\nu$, and we obtain a complete description of the cases in which $E|\langle X, \theta\rangle|^\alpha$ can be guaranteed either to exist or to diverge, just on the basis of $\theta$ and $\nu$.

In the so called secretary problem, if an unknown number $N$ of options arrive at i.i.d. times with a known continuous distribution, then only the geometric, among proper distributions on $N$, has the property that the stopping risk depends just on the elapsed time and not on the number of arrivals so far. But even with such a prior, the optimal rule may, in general, depend on the number of arrivals so far. The optimal rule is closely related to the optimal policy in the Gianini and Samuels infinite secretary problem, except for a linear change in the time scale which depends only on the parameter of the prior, and not on the loss function.

In the context of image analysis, the method of Fourier-domain processing is shown to yield restored signals of optimal quality. This confirms conjectures of statistical optimality that have been made in the past. Quality is measured in terms of convergence rates, and the influence of image smoothness on convergence rates is quantified. This influence is particularly interesting in the case of motion blur, where there is a critical degree of image smoothness (approximately four derivatives of the image) beyond which no improvement in restored image quality may be obtained by passing to smoother images. That is in marked contrast to the case of out-of-focus blur, where restored image quality is always greater for smoother images.

The proof of the existence of a limit distribution for arithmetic multiplicative functions with values in the interval $\lbrack-1, 1\rbrack$, and characterizations of degenerateness and symmetry for such a distribution, can be obtained in a simple manner by combining the famous mean-value theorem of Wirsing with the method of moments of probability theory.