Brownian motion is introduced as a tool in Riemannian geometry to show how useful it is in the function theory of manifolds, as well as the study of maps between manifolds. As applications, a generalization of Picard's little theorem, and a version of it for Riemann surfaces of large genus are given.

Let $X(t), t \geq 0$, be a stationary process assuming values in a measure space $B$. The family of measurable subsets $A_u, u > 0$ is called "rare" if $P(X(0) \in A_u) \rightarrow 0$ for $u \rightarrow \infty$. Put $L_t(u) = \operatorname{mes}\{s: 0 \leq s \leq t, X(s) \in A_u\}$. Under specified conditions it is shown that there exists a function $v = v(u)$ and a nonincreasing function $-\Gamma'(x)$ such that $P(v(u)L_t(u) > x)/E(v(u)L_t(u)) \rightarrow - \Gamma'(x), x > 0$, for $u \rightarrow \infty$ and fixed $t > 0$. If $u = u(t)$ varies appropriately with $t$, then, under suitable conditions, the random variable $v(u)L_t(u)$ has, for $t \rightarrow \infty$, a limiting distribution of the form of a compound Poisson distribution. The results are applied to Markov processes and Gaussian processes.

We study the voter model on $\mathbb{Z}^d, d \geqq 3$, for a sequence $\mu^\varepsilon$ of initial states which have a gradient in the mean magnetization of the order $\varepsilon, \varepsilon \rightarrow 0$. We prove that the magnetization field $m^\varepsilon(f, t) = \varepsilon^d \sum f(\varepsilon x)\eta(x, \varepsilon^{-2}t)$ tends to a deterministic field $m(f, t) = \int dqf(q)m(q, t)$ as $\varepsilon \rightarrow 0. m(q, t)$ is the solution of the diffusion equation. The fluctuations of $m^\varepsilon(f, t)$ around its mean are given by an infinite dimensional, non-homogeneous Ornstein-Uhlenbeck process. In the limit as $\varepsilon \rightarrow 0$, locally, i.e. around $(\varepsilon^{-1}q, \varepsilon^{-2}t)$, the voter model is in equilibrium with parameter $m(q, t)$.

Let $\{\eta^\theta_s(x)\}, s \geq 0, x \in Z^d$ be the basic voter model starting from product measure with density $\theta(0 < \theta < 1).$ We consider the asymptotic behavior, as $t \rightarrow \infty$, of the occupation time field $\{T^x_t\}_{x \in Z^d}$, where $T^x_t = \int^t_0 \eta^\theta_s(x) ds$. Our main result is that, properly scaled and normalized, the occupation time field has a (weak) limit field as $t \rightarrow \infty$, whose covariance structure we compute explicitly. This field is Gaussian in dimensions $d \geq 2$. It is not Gaussian in dimension one, but has an "explicit" representation in terms of a system of coalescing Brownian motions. We also prove that $\lim_{t \rightarrow \infty} T^x_t/t = \theta$ a.s. for $d \geq 2$ (the result is false for $d = 1$). A striking feature of the behavior of the occupation time field is its elaborate dimension dependence.

Let $S$ and $A$ be countable sets and let $\mathscr{G}(\Pi)$ be the set of Markov random fields on $S^A$ (with the $\sigma$-field generated by the finite cylinder sets) corresponding to a specification $\Pi$, Markov with respect to a tree-like neighbour relation in $A$. We define the class $\mathscr{M}(\Pi)$ of Markov chains in $\mathscr{G}(\Pi)$, and generalise results of Spitzer to show that every extreme point of $\mathscr{G}(\Pi)$ belongs to $\mathscr{M}(\Pi)$. We establish a one-to-one correspondence between $\mathscr{M}(\Pi)$ and a set of "entrance laws" associated with $\Pi$. These results are applied to homogeneous Markov specifications on regular infinite trees. In particular for the case $|S| = 2$ we obtain a quick derivation of Spitzer's necessary and sufficient condition for $|\mathscr{G}(\Pi)| = 1$, and further show that if $|\mathscr{M}(\Pi)| > 1$ then $|\mathscr{M}(\Pi)| = \infty$.

Prekopa's theorem gives a qualitative sufficient condition for a completely random point process to be Poisson. A generalization of this theorem is presented. The proof is elementary and uses a combinatorial principle known as Ramsey's theorem.

A certain branching random walk, $\{X_i\}$, on a compact group or a compact homogeneous space is studied. It is proved that the sums $\sum^n_0f(X_i)$ are asymptotically normally distributed for all nice functions $f$ if and only if the Fourier coefficients of the transition probability distribution have real parts not exceeding $\frac{1}{2}$.

Assume that we have a measure $\mu$ on $Sl_2(\mathbf{R})$, the group of $2 \times 2$ real matrices of determinant 1. We look at measures $\mu$ on $Sl_2(\mathbf{R})$ supported on two points, the Bernoulli case. Let $\mathbf{P}^1$ be real projective one-space. We look at stationary measures for $\mu$ on $\mathbf{P}^1$. The major theorem that we prove here gives a general sufficiency condition in the Bernoulli case for the stationary measures to be singular with respect to Haar measure and nowhere atomic. Furthermore, this condition gives the first general examples we know about of continuous singular invariant (stationary) measures of $\mathbf{P}^1$ for measures on $Sl_2(\mathbf{R})$.

Let $X_1, X_2, \cdots$ be a sequence of independent, identically distributed random variables taking values in $\mathbb{R}^d$ and $S_n = X_1 + \cdots + X_n$. For a large class of distributions we obtain estimates for the probability that $S_n$ is in a ball centered at the origin. Such an estimate would follow from a local limit theorem if $X_1$ were in the domain of attraction of a stable law.

Let $X_1, X_2, \cdots$ be a sequence of independent, identically distributed random variables taking values in $\mathbb{R}^d$ and $S_n = X_1 + \cdots + X_n$. For a large class of random variables, which includes all of those in the domain of attraction of a type $A$ stable law, an integral test is given which determines whether $P\{|S_n| \leq \gamma_n \mathrm{i.o.}\} = 0 \quad\text{or}\quad 1$ for any increasing sequence $\{\gamma_n\}$. This result generalizes the Dvoretzky-Erdos test for simple random walk and the Takeuchi and Taylor test for stable random walks.

Khintchine showed that every infinitely divisible law can be obtained as the limit of a subsequence of normed sums of independent, identically distributed random variables. Here we restrict the summands to be in a class which makes the normed sums stochastically compact, i.e. so that every subsequence has a further subsequence which converges to a nondegenerate limit. A nice analytic condition for stochastic compactness was obtained by Feller. Our result is an analogous characterization of the class of limit laws of subsequences of stochastically compact normed sums. One consequence is that they have $C^\infty$ densities.

A transition kernel $(\mu_x)_{x \in X}$ between Polish spaces $X$ and $Y$ is called completely orthogonal if the $\mu_x$ are separated by the fibers of a Borel map $\varphi: Y \rightarrow X$. It is orthogonality preserving if orthogonal measures on $X$ induce orthogonal mixtures on $Y$. We give a von Neumann "type" isomorphism theorem for atomless completely orthogonal kernels, and a theorem and some counterexamples concerning the separation of two orthogonal measure convex sets of probability measures by a measurable set. These techniques yield three results on orthogonality preserving kernels: (1) They need not be completely orthogonal but (2) are uniformly orthogonal (in the sense of D. Maharam) and (3) if $X$ is $\sigma$-compact, $Y = \lim_\leftarrow Y_n$ and $(\mu_x)$ is orthogonality preserving and continuous in $x$ then there is even a strongly consistent sequence of statistics $\varphi_n: Y_n \rightarrow X$ for $(\mu_x)$.

Let $C_\mathscr{I}' = C(\lbrack 0, 1\rbrack; \mathscr{I}')$ be the space of all continuous mappings of $\lbrack 0, 1\rbrack$ to $\mathscr{I}'$, where $\mathscr{L}'$ is the topological dual of the Schwartz space $\mathscr{I}$ of all rapidly decreasing functions. Let $C$ be the Banach space of all continuous functions on $\lbrack 0, 1\rbrack$. For each $\varphi \in \mathscr{I}, \Pi_\varphi$ is defined by $\Pi_\varphi:x \in C_\mathscr{I}' \rightarrow x. (\varphi) \in C$. Given a sequence of probability measures $\{P_n\}$ on $C_\mathscr{I}'$ such that for each $\varphi \in \mathscr{I}, \{P_n\Pi^{-1}_\varphi\}$ is tight in $C$, we prove that $\{P_n\}$ itself is tight in $C_\mathscr{I}'.$ A similar result is proved for the space of all right continuous mappings of $\lbrack 0, 1\rbrack$ to $\mathscr{I}'$.

Consider a stochastically monotone chain with monotone paths on a partially ordered countable set $S$. Let $C$ be an increasing subset of $S$ with finite complement. Then the first passage time from $i \in S$ to $C$ is shown to be IFRA (increasing failure rate on the average). Several applications are presented including coherent systems, shock models, and convolutions of IFRA distributions.

Let $W(T)$ for $0 \leq T < \infty$ be a standard Weiner process and suppose that $c_k$ and $b_k$ are fixed sequences of real numbers satisfying $0 \leq c_k < b_k < \infty$. Let $K(\omega)$ be the set of limit points (as $T \rightarrow \infty$) of $\frac{W(b_k;\omega) - W(c_k;\omega)}{\{2(b_k - c_k)\lbrack\log(b_k/(b_k - c_k)) + \log\log b_k\rbrack\}^{1/2}}$ where $\omega$ is a point in the probability space on which $W(T)$ is defined. We give necessary conditions on $b_k$ and $c_k$ to have $K(\omega) = \lbrack -1, 1\rbrack$ a.s. We also give some related results and discuss sharpness.

Let $S_t$ be a version of the slope at time $t$ of the concave majorant of Brownian motion on $\lbrack 0, \infty)$. It is shown that the process $S = \{1/S_t: t > 0\}$ is the inverse of a pure jump process with independent nonstationary increments and that Brownian motion can be generated by the latter process and Brownian excursions between values of the process at successive jump times. As an application the limiting distribution of the $L_2$-norm of the slope of the concave majorant of the empirical process is derived.

We suggest several Chi squared approximations to the distribution of a sum of independent random variables, and derive asymptotic expansions which show that the error of approximation is of order $n^{-1}$ as $n \rightarrow \infty$. The error may be reduced to $n^{-3/2}$ by making a simple secondary approximation.

A characterization of independence via uncorrelatedness is shown to hold for the families satisfying positive and negative dependence conditions. For the associated random variables, the bounds on covariance functions obtained by Lebowitz (Comm. Math. Phys. $\mathbf{28}$ (1972), 313-321) readily yield such a characterization. An elementary proof for the same characterization is also given for a condition weaker than association, labeled as "strong positive (negative) orthant dependence." This condition is compared with the "linear positive dependence," under which Newman and Wright (Ann. Probab. $\mathbf{9}$ (1981), 671-675) obtained the characterization.

Uniform rates of convergence in the Central Limit Theorem for associated random variables are given. Applications to the Ising Model and Diffusion and Gaussian processes are discussed.

It is shown that if $S_{k,j} = \sum^k_{i = 1} X_{ij}, 1 \leq j \leq d, k \geq 1$ where $(X_{i1}, \cdots, X_{id}), i \geq 1$ are i.i.d. random vectors with positive mean vector $(\mu_1, \cdots, \mu_d)$ and finite covariance matrix $\Sigma$, then for any choice of $\alpha_j$ in $\lbrack 0, 1), 1 \leq j \leq d$ the random vector whose $j$th component is $n^{\alpha_j - 1/2} (\max_{1 \leq k \leq n}S_{k,j}/k^{\alpha_j} - \mu_jn^{1 - \alpha_j})$ converges in law to a multinormal distribution with mean vector zero and covariance matrix $\Sigma$, thereby extending a result of Teicher when $d = 1$.

In this paper we define the expected value of a random vector with respect to a set-valued probability measure. The concepts of independent and identically distributed random vectors are appropriately defined, and a strong law of large numbers is derived in this setting. Finally, an example of a set-valued probability useful in Bayesian inference is provided.