Let $A(1), A(2), \cdots$ be a sequence of independent identically distributed (i.i.d.) random real $n \times n$ matrices and let $x(t) = A(t)x(t - 1), t = 1, 2, \cdots$. Define $\bar{\lambda}_n = \sup\{\lim_{t \uparrow \infty}\|x(t)\|^{1/t}: 0 \neq x(0) \in R^n\}$ where $\|\cdot\|$ denotes, e.g. the Euclidean norm, providing the limit exists almost surely (a.s.) and is nonrandom, and define $\underline\lambda_n$ analogously with sup replaced by inf. If all $n^2$ entries of each $A(t)$ are i.i.d. standard symmetric stable random variables of exponent $\alpha$, then $\underline\lambda_n = \overline\lambda_n = \lambda_n(\alpha)$. In the standard normal case $(\alpha = 2), \lambda_n(2) = (2 \exp\lbrack\psi(n/2)\rbrack)^{1/2}$, where $\psi$ is the digamma function, and $n^{-1/2}\lambda_n(2) \rightarrow 1$; for $0 < \alpha < 2, (n \log n)^{-1/\alpha}\lambda_n(\alpha)$ converges to $\{2 \Gamma(\alpha) \sin(\alpha\pi/2)/\lbrack\alpha\pi\rbrack\}^{1/2}$. Criteria for stability $(\overline\lambda_n < 1)$ and instability $(\underline\lambda_n > 1)$ are investigated for more general distributions of $A(t)$. We obtain, for example, the general bound, $\overline\lambda_n \leq \{r\lbrack E(A(1)^T A(1))\rbrack\}^{1/2}$, where $A^T$ is the transpose of $A$ and $r$ denotes the spectral radius. In the case of independent entries of mean zero and common variance $s^2/n$, this leads to $\lim \sup_n \overline\lambda_n \leq s$. If the entries of $A(t)$ are i.i.d. and distributed as $W/n^{1/2}$ where $W$ is independent of $n$, has mean zero, variance $s^2$ and satisfies $E(\exp\lbrack iuW\rbrack) = O(|u|^{-\delta})$ as $|u| \uparrow \infty$ for some $\delta > 0$, then $\lim \inf_n\bar\lambda_n \geq s$. These conditions for the asymptotic stability or instability of a product of random matrices are of the form originally proposed by May for differential equations governed by a single random matrix. We give counterexamples to show that May's criteria for the system of linear ordinary differential equations that he considered are not valid in the generality originally proposed, nor are they valid for the related system of difference equations considered by Hastings. The validity of May's criteria for these systems under more restrictive hypotheses remains an open question.

Considering difference equations in discrete space instead of differential equations in Euclidean space, we investigate a probabilistic formula for the solution of the Dirichlet problem for biharmonic functions. This formula involves the expectation of a weighted sum of the pay-offs at the successive times at which the Markov chain is in the complement of the domain. To make the infinite sum converge, we use Borel's summability method. This is interpreted probabilistically by imbedding the Markov chain into a continuous time, discrete space Markov process.

We first recall the concept of hydrodynamical equation of an infinite particle system. We then prove that the equation associated to the zero-range process is a first order nonlinear partial differential equation.

Using a simple symmetrization procedure, an upper bound is obtained for the probability distribution of various kinds of weighted empirical distribution processes where the underlying real valued random variables are not identically distributed. These probability bounds are used to obtain bounded laws of the iterated logarithm for empirical processes with different kinds of weighting. They are also used to obtain a one sided version of Daniel's theorem in the non-i.i.d. case.

We derive sharp finite sample estimates and exact almost sure limit results for local deviations of multivariate empirical processes. These are useful for obtaining, e.g., exact convergence rates of multivariate kernel density estimators. It is also indicated how local properties of multivariate empirical processes may be used to study various problems in nonparametric multivariate analysis.

Let $X_1, X_2, \cdots$ be i.i.d. random variables and let $M_n = \max_{i \leq j \leq n} X_j$. For each real sequence $\{b_n\}$, a sequence $\{b^\ast_n\}$ and a sub-sequence of integers $\{n_k\}$ is explicitly constructed such that $P(M_n \leq b_n \text{i.o.}) = 1 \operatorname{iff} \sum_k P(M_{n_k} \leq b^\ast_{n_k}) = \infty$. This result gives a complete characterization of the upper and lower-class sequences (as introduced by Paul Levy) for the a.s. minimal growth rate of $\{M_n\}$.

This paper deals with the convergence in distribution to Gaussian, generalized Poisson and infinitely divisible laws of the row sums of certain $\phi$ or $\psi$-mixing triangular arrays of Banach space valued random vectors with stationary rows. Necessary and sufficient conditions for convergence in terms of individual r.v.'s are proved. These include sufficient conditions for the convergence to a stable law of the normalized sums of certain $\phi$-mixing, stationary sequences. An invariance principle for stationary, $\phi$-mixing triangular arrays is given.

Probabilities of large deviations for sums of i.i.d. Banach space valued random variables are investigated when the laws of the random variables converge weakly and a uniform exponential integrability condition is satisfied. Furthermore, a discussion of possible improvements of the estimates is given, when the probability is estimated that the sum lies in a convex set.

We find exact upper and lower bounds for the distribution of the supremum of a homogeneous Gaussian random field with pyramidal covariance function. The upper bound comes from a reflection principle type argument. The lower bound is found by exploiting a relationship between this random field and a particular Banach space valued process in one-dimensional time.

Let $M = \{M(z), z \in \lbrack 0, 1\rbrack^2\}$ be a two-parameter square integrable continuous martingale. We prove the sample continuity of the quadratic variation of $M$ using an Ito's differentiation formula for $M^2$.

It is shown that certain measure-valued stochastic processes describing the time evolution of systems of weakly interacting particles converge in the limit, when the particle number goes to infinity, to a deterministic nonlinear process.

Let $\{T_n\}_{n \geq 1}$ be a sequence of independent and identically distributed strongly continuous semigroups on a separable Banach space. The corresponding generators $\{A_n\}_{n \geq 1}$ satisfy $E\lbrack A_n\rbrack = 0$. Conditions are given to guarantee that the weak limit $Y(t) = \text{limit}_{n \rightarrow \infty} \prod^{\lbrack n^2t\rbrack}_{i = 1} T_i(1/n) Y_n(0)$ exists, and is characterized as the unique solution of a martingale problem. Transport phenomena, random classical mechanics, and families of bounded operators are the featured examples.

We prove a central limit theorem for families $\{X_n(N): \mathbf{n} \in \Lambda(N)\}$ of associated random variables indexed by subsets $\Lambda(N)$ of $\mathbb{Z}^d$, as $N \rightarrow \infty$; this is an extension of the Newman-Wright invariance principle for associated stationary sequences $\{X_n: n \geq 1\}$ satisfying $\sum_n \operatorname{cov}(X_1, X_n) < \infty$, but with the stationarity property replaced by conditions on the moments of the $X$'s. The theorem has applications to the voter model and the percolation model. In the latter case, it provides an extension of a central limit theorem of the authors [4], by reducing the severity of the moment conditions. Also, we prove a central limit theorem for certain non-stationary non-associated families of random variables which arise in percolation theory. This includes, for example, a central limit theorem for the number of open clusters contained within the circuit $\gamma(n)$ of $\mathbb{Z}^2$, where $\{\gamma(n)\}$ is a sequence of circuits which satisfy a regularity condition and whose interiors $\{\overset{\circ}\gamma(n)\}$ satisfy $|\overset{\circ}\gamma(n)| \rightarrow \infty$ as $n \rightarrow \infty$.

Oseledec's Multiplicative Ergodic Theorem is used to give recurrence and transience criteria for random walk in a random environment on the integers. These criteria generalize those given by Solomon in the nearest-neighbor case. The methodology for random environments is then applied to Markov chains with periodic transition functions to obtain recurrence and transience criteria for these processes as well.

Suppose $\{X_1, X_2, \cdots\}$ are i.i.d. random variables with finite mean $0 < E(X_1) < \infty$. If $S_n$ stands for the $n$th partial sum, and $\{a(n)\}_n$ is a sequence of nonnegative numbers, then $G(x) = \sum^\infty_{n = 0} a(n)P\{S_n \leq x\}$ is a generalized renewal measure. We investigate the behaviour of $G(x + h) - G(x)$ as $x \rightarrow \infty$ for $\{a(n)\}_n$ regularly varying.

Gnedenko's (1943) study of the class $\mathscr{G}$ of limit laws for the sequence of maxima $M_n \equiv \max\{X_0, \cdots, X_{n - 1}\}$ of independent identically distributed random variables $X_0, X_1, \cdots$ is extended to limit laws for weighted sequences $\{w_n(\gamma)X_n\}$ (the simplest case $\{\gamma^nX_n\}$ has geometric weights $(0 \leq \gamma < 1))$ and translated sequences $\{X_n - v_n(\delta)\}$ (the simplest case is $\{X_n - n\delta\} (\delta > 0))$. Limit laws for these simplest cases belong to the family $\mathscr{G}$ characterized by Gnedenko; with more general weights or translates, limit laws outside $\mathscr{G}$ may arise.

Necessary and sufficient conditions are given under which all partial limit distributions for properly normalized sample extremes of i.i.d. random variables are proper and nondegenerate. In the process we study a new type of extended regular variation called asymptotic balance that should be useful in other contexts as well.

It is shown that a necessary and sufficient condition, for the existence of a mean-convergent series for the linear interpolator of a $q$-variate stationary stochastic process $\{X_n\}$ with density matrix $W$, is that the Fourier series of the isomorph of the linear interpolator should converge in the norm of $L^2(W)$, and this happens if the past and future of the process are at positive angle. This provides a positive answer to a question of H. Salehi (1979) concerning the square summability of the inverse of $W$ and improves upon the work of Rozanov (1960) and Salehi (1979).

We consider finitely additive random measures taking independent values on disjoint Borel sets in $R^k$, and ask when such measures, restricted to some subclass $\mathscr{A}$ of closed Borel sets, possess versions which are "right continuous with left limits", in an appropriate sense. The answer involves a delicate relationship between the "Levy measure" of the random measure and the size of $\mathscr{A}$, as measured via an entropy condition. Examples involving stable measures, Dudley's class $I(k, \alpha, M)$ of sets in $R^k$ with $\alpha$-times differentiable boundaries, and convex sets are considered as special cases, and an example given to show what can go wrong when the entropy of $\mathscr{A}$ is too large.

In this note we prove a generalisation of a conditioned functional central limit theorem of Bolthausen (cf. [3]). This generalisation explains the nature of the discontinuity between such conditioned limit theorems for random walks (or in queueing for waiting times) with or without drift.

It is shown that a method recently developed by Bolthausen permits an extension (up to a logarithmic factor) of an estimate of the rate of convergence in the martingale central limit theorem due to Heyde and Brown.

In the note a new class of functions called $\alpha$-quasi-concave is introduced and it is proved that the convolution of two unimodal functions each having some additional concavity properties belongs to this class. Other results concerning the convolution of unimodal functions are also studied and the extensions of some of them are also given in the paper.