A large class of generalized random fields is defined, containing random elements $F$ of $\mathscr{J}'$, where $\mathscr{J}'$ is the dual of the Schwartz space $\mathscr{J} = \mathscr{J}(\mathbb{R}^\nu)$. Such a generalized random field is translation-invariant if $F\phi$ is the same as $F\psi$ for any translate $\psi$ of $\phi$; it is invariant under the renormalization group with index $_\kappa$ (or self-similar with index $_\kappa$) if $F\phi_\lambda = \lambda^{-\alpha}F\phi$ for all $\lambda > 0$ and $\phi \in \mathscr{L}$, where $\phi_\lambda$ is the rescaled test function $\phi_\lambda(x) = \lambda^{-\nu}\phi(x/\lambda)$. Recent work of several authors has shown that self-similar generalized random fields on $\mathbb{R}^\nu$, and self-similar random fields on $\mathbb{Z}^\nu$ which can be constructed from them, arise naturally in problems of statistical mechanics and limit laws of probability theory. They generalize the theory of stable distributions. Here the class of all translation-invariant self-similar Gaussian generalized random fields on $\mathbb{R}^\nu$ is completely described. By the discretization of such fields the class of self-similar Gaussian fields with discrete arguments (found by Sinai) is extended. Finally, a class of generalized random fields subordinated to the self-similar translation-invariant Gaussian ones is constructed. These non-Gaussian generalized random fields are Wick powers (multiple Ito integrals) of the Gaussian ones.

In this paper we prove approximation theorems of the following type. Let $\{X_k, k \geqslant 1\}$ be a sequence of random variables with values in $\mathbb{R}^{d_k}, d_k \geqslant 1$ and let $\{G_k, k \geqslant 1\}$ be a sequence of probability distributions on $\mathbb{R}^{d_k}$ with characteristic functions $g_k$ respectively. If for each $k \geqslant 1$ the conditional characteristic function of $X_k$ given $X_1, \cdots, X_{k - 1}$ is close to $g_k$ and if $G_k$ has small tails, then there exists a sequence of independent random variables $Y_k$ with distribution $G_k$ such that $|X_k - Y_k|$ is small with large probability. As an application we prove almost sure invariance principles for sums of independent identically distributed random variables with values in $\mathbb{R}^d$ and for sums of $\phi$-mixing random variables with a logarithmic mixing rate.

Let a distribution function $F(x), \int xdF(x) = 0, \int x^2dF(x) = 1$ be given. Strassen constructed two sequences $X_1, X_2, \ldots$ and $Y_1, Y_2, \ldots$ of independent, identically distributed random variables, the $X_i$ with distribution function $F(x)$ and the $Y_i$ with standard normal distribution, in such a way that the partial sums $S_n = \sum^n_{i = 1}X_i$ and $T_n = \sum^n_{i = 1} Y_i$ satisfy the relation $|S_n - T_n| = O((n \log \log n)^\frac{1}{2})$ with probability 1. Earlier we proved that this result cannot be improved. Now we show however that an approximation $|S_n - T_n| = O(n^\frac{1}{2})$ can be achieved, if the $Y_i$ are independent normal variables whose variances are appropriately chosen.

Vector-valued random processes, $\mathbf{X}(t)$, can be "enveloped" by set-valued random processes, $\mathscr{S}(t)$, to which they belong with probability 1 during any finite length of time. When applied to scalar processes, the set-definition of envelope includes and is richer than the familiar point-definitions. Several random set-envelope processes in $n$-dimensional space, $R_n$, are defined and the mean rates at which they "cross" given regions of $R_n$ are calculated. Comparison is made with the mean crossing rates of associated enveloped Gaussian processes, $\mathbf{X}(t)$.

This paper explores the strong law of large numbers in the infinite dimensional setting. It is shown that under several classical conditions--such as the Kolmogorov condition--the strong law holds if and only if the weak law holds.

Laws of large numbers are obtained for random variables taking their values in $D\lbrack 0, 1\rbrack$ where $D\lbrack 0, 1\rbrack$ is equipped with the Skorokhod topology. The strong law of large numbers is obtained for independent, convex tight random elements $\{X_n\}$ satisfying $\sup_nE\|X_n\|^r_\infty < \infty$ for some $r > 1$ where $\|X\|_\infty = \sup_{0 \leqslant t \leqslant 1}|x(t)|$. A strong law of large numbers is also obtained for almost surely monotone random elements in $D\lbrack 0, 1\rbrack$ for which the hypothesis of convex tightness is not needed. A discussion of the condition of convex tightness is also included.

The relationship of the large deviation rate, $\psi^\ast(a)$, of the mean of independent and identically distributed random variables to their cumulant generating function, $\psi(\lambda)$, is well known. This paper studies how the behavior of the sign changes of $\psi_1^\ast(a) - \psi_2^\ast(a)$ is related to that of $\psi_1(\lambda) - \psi_2(\lambda)$ for cumulant generating functions $\psi_1$ and $\psi_2$ with rates $\psi_1^\ast$ and $\psi_2^\ast$, respectively. Use is made of the fact that the rate $\psi^\ast$ is nothing more than the conjugate convex function of $\psi$. Results concerning the relationship of the behavior of the difference of convex functions to that of the difference of their conjugates are first proven and then applied to determine the relationship of the behavior of the sign changes of $\psi_1^\ast - \psi_2^\ast$ to that of $\psi_1 - \psi_2$. Results are also given relating this behavior to that of $F_1 - F_2$ and $f_1 - f_2$, where $F_i$ and $f_i(i = 1$ and 2) are the distribution function and the density function, respectively, corresponding to $\psi_i$.

In studying the asymptotic properties of certain isotone regression estimators, one is led to consider the maximum of sums of independent random variables indexed by a partially ordered set. An index set which is a sequence of $\beta$ dimensional vectors, $\{t_k\}^\infty_{k = 1}$, and the usual partial order on $R_\beta$, the $\beta$ dimensional reals, are considered here. The random variables are assumed to satisfy a condition equivalent to a finite first moment in the identically distributed case and are assumed to be centered at their means. For $A \subset R_\beta$, let $S_n(A)$ denote the sum of those random variables with indices $t_k \in A$ and $k \leqslant n$. It is shown that if the sequence $\{t_k\}$ satisfies a certain condition, then the maximum, over all upper layers $U$ in $R_\beta$, of $S_n(U)/n$ converges almost surely to zero. As a corollary to this result one obtains the strong consistency of this isotone regression estimator. If the sequence $\{t_k\}$ is a realization of a sequence of independent, identically distributed, $\beta$ dimensional random vectors and if the probability induced by such a vector is discrete, absolutely continuous or a mixture of the two, then the condition on the sequence $\{t_k\}$ is satisfied almost surely. Some nondiscrete, singular induced probabilities of interest in these regression problems are considered also.

As yet no efficiently computable algorithm for one step nonlinear prediction has been proposed for any general class of stationary processes which performs strictly better than the optimal linear predictor. In this paper it is shown that for the class of stationary moving average processes the improvement obtained by optimal nonlinear prediction versus optimal linear prediction is bounded by a constant which depends only on the distribution of the independent and identically distributed random variables $Y_j$ used to form the moving average process $X_n = \sum a_jY_{n - j}$.

We prove a complete pointwise ergodic theorem for Harris' basic one-dimensional two-sided nearest neighbor contact process, provided that the corresponding one-sided contact process is nonergodic.

It is shown that Brownian excursion is equal in distribution to Brownian bridge with the origin placed at its absolute minimum. This explains why the maximum of Brownian excursion and the range of Brownian bridge have the same distribution, a fact which was discovered by Chung and Kennedy. The result is proved by establishing similar relations for "Bernoulli excursions" and "Bernoulli bridges" constructed from symmetric Bernoulli walks, and exploiting known weak convergence results. Some technical complications arise from the fact that Bernoulli bridges assume their minimum value with positive probability more than once.

A strong law of large numbers is proved for tight, independent random elements (in a separable normed linear space) which have uniformly bounded $p$th moments $(p > 1)$. In addition, a weak law of large numbers is obtained for tight random elements with uniformly bounded $p$th moments $(p > 1)$ where convergence in probability for the separable normed linear space holds if and only if convergence in probability for the weak linear topology holds.

In 1967, Komlos proved that if $\{\xi_n\}$ is a sequence of real random variables for which $\sup_{n \geqslant 1}E|\xi_n| < \infty$, then there exists a subsequence $\{\eta_n\}$ of $\{\xi_n\}$ and an integrable random variable $\eta$ such that for an arbitrary subsequence $\{\eta_n\}$ of $\{\eta_n\}$. $$\lim_{n \rightarrow \infty} \frac{1}{n}(\check{\eta}_1 + \check{\eta}_2 + \cdots + \check{\eta}_n) = \eta \mathrm{a.s.}$$ In this paper, we attempt to extend this result to separable Banach space valued random elements. We impose a condition stronger than uniform integrability.

Let $X$ and $Y$ be random variables and assume $X$ has a density $f_X(x)$. An inversion theorem for the conditional expectation $E(Y\mid X = x)$ is derived which generalizes and simplifies that of Yeh. As an immediate corollary an almost-sure version of Bartlett's formula for the conditional characteristic function of $Y$ given $X = x$ is obtained. This result is applied to show the existence under regularity conditions of a version of the regular conditional distribution $P\{dy\mid X = x\}$ which is well defined for those values of $x$ such that $f_X(x) \neq 0$.

Let $P$ be a probability measure defined on a $\sigma$-field $\mathscr{F}$ over $\Omega$. Let $\mathfrak{L}\subset \mathscr{F}$ be a $\sigma$-lattice and $r > 1$. For each $A \in \mathscr{F}$ denote by $P_r(A/\mathfrak{L})$ the unique nearest point projection of $1_A$ onto the closed convex subspace of all "$\mathfrak{L}$-measurable" equivalence-classes of $L_r(\Omega, \mathscr{F}, P)$. It is shown that there exists a functional relationship between $P_r(A/\mathfrak{L})$ and $P_2(A/\mathfrak{L})$ of the form $$P_r(A/\mathfrak{L}) = \varphi(P_2(A/\mathfrak{L}))$$ where the function $\varphi$ depends only on $r$ but not on $A, P$ or $\mathfrak{L}$. This relationship is applied to the theory of sufficiency.

It is shown that the only stationary countable state Markov chains satisfying a mixing condition referred to as "of Markov type" are sequences of independent random variables.

The range of almost sure limits of $F$-variation for a class of Gaussian random fields is considered by adopting a class of sequences of partitions in the parameter space of the random field. The application to Levy's Brownian motion explains, in the case of two-dimensional parameters, that the almost sure limit given by Berman is the maximum in a range.

Let $(\Omega, \mathscr{A}, \mu)$ be a probability space and let $L$ be an ideal in $M(\Omega, \mathscr{A}, \mu)$ containing $\chi_\Omega$. A one-one correspondence between the class of "order closed" linear subspaces of $L$ and the sub $\sigma$-algebras of $\mathscr{A}$ is proved. Furthermore, if $T : L \rightarrow M(\Omega, \mathscr{A}, \mu)$ is a strictly positive order continuous projectionlike linear map then $T$ is shown to be a conditional expectation $E_\nu(\cdot \mid\mathscr{A}_0)$. It follows that if $T: L \rightarrow M(\Omega, \mathscr{A}, \mu)$ is a positive expectation invariant projectionlike linear map, then even $T = E_\mu(\cdot \mid \mathscr{A}_0)$.

An extension is derived of a $p$-function inequality due to Blackwell and Freedman.

Let $B$ be a standard $n$-dimensional Brownian motion, let $A$ be compact and let $\nu$ be a probability measure on $\partial A$. We treat the following inverse exit problem: describe the set $M(\nu)$ of all probability measures $\mu$ on $A$ such that $P^\mu\{B(T)\in \cdot\} = \nu(\cdot)$, where $T$ is the time of first exit from $A$. Elements of $M(\nu)$ are characterized in terms of integrals of harmonic functions with respect to them. For $n = 1$, extreme points of $M(\nu)$ are computed in closed form; for $n \geqslant 2$, extreme points of $M(\nu)$ are characterized. Geophysical and potential-theoretic aspects of the problem are discussed.