An analysis of Wiener functionals is studied as a kind of Schwartz distribution theory on Wiener space. For this, we introduce, besides ordinary $L_p$-spaces of Wiener functionals, Sobolev-type spaces of (generalized) Wiener functionals. Any Schwartz distribution on $\mathbf{R}^d$ is pulled back to a generalized Wiener functional by a $d$-dimensional Wiener map which is smooth and nondegenerate in the sense of Malliavin. As applications, we construct a heat kernel (i.e., the fundamental solution of a heat equation) by a generalized expectation of the Dirac delta function pulled back by an Ito map, i.e., a Wiener map obtained by solving Ito's stochastic differential equations. Short-time asymptotics of heat kernels are studied through the asymptotics, in terms of Sobolev norms, of the generalized Wiener functional under the expectation.

Let $\mu^n$ be the distribution of a product of $n$ independent identically distributed random matrices. We study tightness and convergence of the sequence $\{\mu^n, n \geq 1\}$. We apply this to linear stochastic differential (and difference) equations, characterize the stability in probability, in the sense of Hashminski, of the zero solution, and find all their stationary solutions.

Let $X_1, X_2,\ldots$ be nonnegative i.i.d. random variables and $S_n = X_1 + \cdots + X_n; EX_1 = \mu \leq \infty$ and $a$ is the infimum of the support of the distribution of $X_1$. For $a < x_n < \mu$ we obtain the asymptotic behavior of $\log P\{S_n \leq nx_n\}$ as $n \rightarrow \infty$. Under the additional assumption of stochastic compactness a stronger result is obtained which gives the asymptotic behavior of $P\{S_n \leq nx_n\}$ itself. Analogues of these results are given for subordinators when $t \rightarrow \infty$ or $t \rightarrow 0$.

Let $S_n, n \geq 1$, be a random walk and $t = t_a = \inf\{n \geq 1: ng(S_n/n) > a\}$. The main results of this paper are two-term asymptotic expansions as $a \rightarrow \infty$ for the marginal distributions of $t_a$ and the normalized partial sum $S^\ast_t = (S_t - t\mu)/\sigma\sqrt t$. To leading order, $S^\ast_t$ has a standard normal distribution. The effect of the randomness in the sample size $t$ on the distribution of $S^\ast_t$ appears in the correction term of the expansion.

We are concerned with the stationary distribution of reflected Brownian motion (RBM) in a $d$-dimensional domain $G$. Such a process behaves like Brownian motion with a constant drift vector $\mu$ in $G$ and is instantaneously reflected at the boundary, with the direction of reflection given by a non-tangential vector field $\nu$ on $\partial G$. We consider first the case where $G$ is smooth and bounded and $\nu$ varies smoothly over $\partial G$. It is shown that the RBM has a stationary density of the exponential form $C(\mu)\exp\{\gamma(\mu) \cdot x\}$ for each $\mu \in R^d$ if and only if $\nu$ satisfies a certain skew symmetry condition. An explicit formula is given for $\gamma(\mu)$ in terms of $\nu$ and $\mu$. Motivated by applications in queueing theory, we next consider the case where $G$ is a convex polyhedral domain and $\nu$ is constant on each face of the boundary. Postponing for now the treatment of certain foundational questions, we work directly with a basic adjoint relation (BAR) that appears to characterize stationary distributions for a wide class of RBM's in polyhedral domains. This analytic relation is motivated by formal analogy with the smooth case and will be rigorously justified in later work. As in the smooth case, it is found that (BAR) has a solution of exponential form for each $\mu \in R^d$ if and only if $\nu$ satisfies a certain skew symmetry condition. Moreover, under a mild nondegeneracy condition, it is shown that an exponential solution exists for one $\mu \in R^d$ if and only if such a solution exists for every $\mu \in R^d$.

If $\{X_n, n \geq 1\}$ is an exchangeable sequence with $(1/b_n(\sum^n_1X_i - a_n)) \rightarrow N(0, 1)$ for some constants $a_n$ and $0 < b_n \rightarrow \infty$ then $b_n/n^\alpha$ is slowly varying with $\alpha = 1$ or $\frac{1}{2}$ and necessary conditions (depending on $\alpha$) which are also sufficient, are obtained. Three such examples are given, one with infinite mean, one with no positive moments, and the third with almost all conditional distributions belonging to no domain of attraction of any law.

This note provides necessary and sufficient conditions for the law of large numbers for partial sum processes indexed by sets of $\lbrack 0, 1\rbrack^d$ and based on i.i.d. integrable random variables.

If $f = \{f_t\mid t \in T\}$ is a centered, second-order stochastic process with bounded sample paths, it is then known that $f$ satisfies the central limit theorem in the topology of uniform convergence if and only if the intrinsic metric $\rho^2_f$ (on $T$) induced by $f$ is totally bounded and the normalized sums are eventually uniformly $\rho^2_f$-equicontinuous. We show that a centered, second-order stochastic process satisfies the central limit theorem in the topology of uniform convergence if and only if it has bounded sample paths and there exists totally bounded pseudometric $\rho$ on $T$ so that the normalized sums are eventually uniformly $\rho$-equicontinuous.

Sufficient conditions, which under weak additional assumptions are also necessary, are found for the central limit theorem for empirical processes indexed by classes of functions which have the Vapnik-Cervonenkis property. This improves an earlier theorem of Pollard (1982a), and leads to necessary and sufficient conditions for the CLT for weighted empirical processes indexed by Vapnik-Cervonenkis classes of sets.

To a class $\mathscr{F}$ of bounded functions on a probability space we associate two classes $\mathscr{F}_r$ and $\mathscr{F}_s$. The class $\mathscr{F}$ is a Donsker class if and only if $\mathscr{F}_r$ and $\mathscr{F}_s$ are Donsker classes. The class $\mathscr{F}_r$ corresponds to a separable version of the empirical process. It is obtained by applying a special type of lifting to $\mathscr{F}$. The class $\mathscr{F}_s$ consists of positive functions that are zero almost surely. It concentrates the pathology of $\mathscr{F}$ with respect to measurability. We use this method to prove without any measurability assumption a general contraction principle for processes that satisfy the central limit theorem.

The strong Markov property of a process $X$ at a stopping time $\tau$ may be split into a conditional independence part (CI) and a homogeneity part (H). However, it turns out that (H) often implies at least some version of (CI). In the present paper, we shall assume that (H) holds on the set $\{X_\tau \in B\}$, for all stopping times $\tau$ such that $X_\tau \in F$ a.s., where $F$ is a closed recurrent subset of the state space $S$, while $B \subset F$. If $F = S$, then (CI) will hold on $\{X_\tau \in B\}$ for every stopping time $\tau$, so in this case $X$ is regenerative in $B$. In the general case, the same statement is conditionally true in a suitable sense, given some shift invariant $\sigma$-field.

A class of standard processes which admit local times at each point is considered. The following regularity properties are assumed: $T_x \rightarrow T_a = 0$ (as $x \rightarrow a$) in $P^a$-probability and $P^a(T_b < \infty) > 0$ for all pairs of points $a, b (T_x = \inf\{t > 0: X_t = x\})$. The class under consideration turns out to be very large. It is already known that a wide class of processes with independent increments fulfill our hypothesis. We also observe that the class is left invariant by the usual transformations: time change, subprocess and $u$-process ($h$-path) transformations. The first important result of the paper is that every continuous additive functional may be represented as a mixture (integral) of local times. This theorem is used to prove two further results. The first one asserts that every process in the class has a dual process which remains in the class. Particularly Hunt's hypothesis (F) is satisfied. The second one generalises the occupation time and downcrossing approximating models. Such approximation theorems are proved for a C.A.F. whose representing measure is given.

Using the continuum hypothesis, we produce an example answering three problems from the theory of right processes. In particular, we give a nontrivial example of a strong Markov process which is not a right process.

A new proof of a sharp $L^p$-inequality for stochastic integrals is given that makes it possible to show that strict inequality holds in all nontrivial cases.

Take $n$ points at random in a fixed set in $R^d$. Define the maximal spacing, e.g., as the volume of the largest ball that is contained in the fixed set and avoids all $n$ chosen points. The asymptotic distribution of the maximal spacing and strong bounds are given.

Let $(X_n)_{n=1,2,\ldots}$ be a strictly stationary sequence of real-valued random variables. Let $M_{i,j} = \max(X_{i+1},\ldots, X_j)$ and let $M_n = M_{0,n}$. Let $(c_n)$ be a sequence of real numbers. It is shown under general circumstances that $P\lbrack M_n \leq c_n\rbrack - (P\lbrack X_1 \leq c_n\rbrack)^{nP\lbrack M_{1,p_n}\leq c_n\mid X_1>c_n\rbrack} \rightarrow 0$, for any sequence $(p_n)$ satisfying certain growth-rate conditions. Under suitable mixing conditions, there exists a distribution function $G$ such that $P\lbrack M_n \leq c_n\rbrack - (G(c_n))^n \rightarrow 0$ for all sequences $(c_n)$. These theorems hold in particular if $(X_n)$ is a function of a positive Harris Markov sequence. Some examples are included.

We prove the following conjecture of Dudley: If the volume exponent of a compact convex and symmetric subset $K$ of a Hilbert space is less than $-1$, then $K$ is a G.C. set.

We study the asymptotic behaviour of the solution of the stochastic difference equation $X_{n+1} = X_n + g(X_n)(1 + \xi_{n+1})$, where $g$ is a positive function, $(\xi_n)$ is a 0-mean, square-integrable martingale difference sequence, and the states $X_n < 0$ are assumed to be absorbing. We clarify, under which conditions $X_n$ diverges with positive probability, satisfies a law of large numbers, and, properly normalized, converges in distribution. Controlled Galton-Watson processes furnish examples for the processes under consideration.

Sufficient conditions for the existence of a limiting distribution for a multitype branching process with immigration in a random environment, $Z(t)$, are given. In the case when the environment is an independent, identically distributed sequence, sufficient conditions are given which insure that the tail of the distribution of $\nu = \inf\{t > 0: Z(t) = 0\}$ decreases exponentially fast, and an application of this fact to random walk in a random environment is indicated.

Recently van den Berg and Kesten have obtained a correlation-like inequality for Bernoulli sequences. This inequality, which goes in the opposite direction of the FKG inequality, states that the probability that two monotone (i.e., increasing or decreasing) events "occur disjointly" is smaller than the product of the individual probabilities. They conjecture that the monotonicity condition is immaterial, i.e., that the inequality holds for all events. In the present paper we try to make clear the intuitive meaning of the conjecture and prove some nontrivial special cases, one of which, a pure correlation inequality, is an extension of Harris' FKG inequality.

We prove that the position of a tagged particle in a stationary simple exclusion process satisfies a law of large numbers. For this purpose, we show the extremality of an invariant measure for the process "seen" from the tagged particle, and we use the ergodicity properties of the initial process.

A new proof is given of the complete convergence theorem for the $d$-dimensional basic contact process provided that the infection parameter is larger than the critical value in the one-dimensional case. This proof is much more elementary than the known one since it does not depend on exponential estimates and does not use the subadditive ergodic theory in the extension from one to more dimensions.

Let $F(\cdot)$ be a c.d.f. on $(0, \infty)$ such that $1 - F(x)$ is regularly varying with exponent $-\alpha, \frac{1}{2} < \alpha \leq 1$. Let $Q(\cdot): \mathscr{R}^+ \rightarrow \mathscr{R}^+$ be nonincreasing and regularly varying with exponent $-\beta, 0 \leq \beta < 1$. Then, as $t \rightarrow \infty, (U \ast Q)(t) \equiv \int_{\lbrack 0,t\rbrack}Q(t - u)U(du)$ is asymptotic to $c(\alpha, \beta)(\int^t_0Q(u) du)(\int^t_0(1 - F(u)) du)^{-1}$, where $U(\cdot)$ is the renewal function associated with $F(\cdot)$ and $c(\alpha, \beta)$ is a suitable constant. This is an improved version of a theorem due to Teugels, whose proof appears to be incomplete. Applications of the result to the second order behavior of $U(t)$ in some special cases are also given.

Let $(X_k)_{k\in\mathbb{N}}$ be a sequence of i.i.d. random variables with partial sums $S_0 = 0, S_n = \sum^n_{k=1} X_k$. We investigate the behaviour of $\sum^\infty_{n=0} a_nP(S_n \in x + A)$ as $x \rightarrow \pm \infty$, where $(a_n)_{n\in\mathbb{N}_0}$ is a sequence of nonnegative numbers and $A \subset \mathbb{R}$ is a fixed Borel set.

It is known that the set of correlation coefficients formed from $k$ independent normal samples exhibits pairwise independence of its members (Geisser and Mantel (1962)). Here it is shown that many much larger subsets of the matrix are fully independent. The main result characterises such subsets in a simple way. Because the results are framed in abstract terms, they also apply to rank correlation coefficients and $\chi^2$ statistics.

The class of nonnegative random vectors $\mathbf{T} = (T_1,\ldots, T_n)$ for which $\min_{1\leq i\leq n} a_iT_i$ is IFRA for all $0 < a_i \leq \infty, i = 1,\ldots, n$, is closed under convolution.