We study the behavior of an interacting particle system known as the voter model in two dimensions. This process provides a simple example of "critical clustering" among two colors, say green and black, in the plane. The paper begins with some computer simulations, and a survey of known results concerning the voter model in the three qualitatively distinct cases: three or more dimensions (high), one dimension (low), and two dimensions (critical). Our main theorem, for the planar model, states roughly that at large times $t$ the proportion of green sites on a box of side $t^{\alpha/2}$ centered at the origin fluctuates with $\alpha$ according to a time change of the Fisher-Wright diffusion. Some applications of the theorem, and several related results, are described.

A duality theory is developed which works for general Markovian spin-flip systems with attractive rates. This theory is applied to one-dimensional nearest neighbor translation invariant systems to extend results which were first proved for the contact process by Durrett and Griffeath (1983). In particular, exponential convergence to equilibrium starting from all 1's is shown for noncritical nonergodic systems (Theorem 2). As a consequence, two different definitions of the critical value are shown to be equivalent (Theorem 5). In the course of the proof of Theorem 2, a new result concerning the distribution of the system near edges is obtained (Theorem 4).

We prove that a tagged particle in asymmetric simple exclusion satisfies a central limit theorem when properly rescaled. To obtain this result we derive several results of positive (or negative) correlation for occupation times of a server in a series of queues which imply various central limit theorems.

We consider stationary, reversible exclusion processes with speed change and prove that for sufficiently small interaction the fluctuation fields constructed from local functions become proportional to the density fluctuation field when averaged over suitably large space-time regions. If the exclusion process is of gradient type, this result implies that the density fluctuation field converges to an infinite dimensional Ornstein-Uhlenbeck process.

A mosaic process is formed by centering independent and identically distributed random shapes at the points of a Poisson process in $k$-dimensional space. Clusters of overlapping shapes are called clumps. This paper provides approximations to the distribution of the number of clumps of a specified order within a large region. The approximations cover two different situations--"moderate-intensity" mosaics, in which the covered proportion of the region is neither very large nor very small; and "sparse" mosaics, in which the covered proportion is quite small. Both these mosaic types can be used to model observed phenomena, such as counts of bacterial colonies in a petri dish or dust particles on a membrane filter.

Let $S$ be a countable set, and $a, b$ two distinct elements not in $S$. Let $\Omega$ be the set of functions $\omega: Z$ (integers) $\rightarrow S \cup \{a, b\}$ so that (i) $\omega (n) \in S$ for some $n$, (ii) $\omega (n) = a \Rightarrow \omega(m) = a$ for $m < n$, and (iii) $\omega(n) = b \Rightarrow \omega(m) = b$ for $m > n$. On $\Omega$, let $x(n)$ be the $n$th coordinate, $\theta_n$ the shift operator, and $\alpha = \inf\{n: x(n) \in S\}$. A measure $P$ on $\Omega$ is Markov if $\forall i \in S, n \in Z, P(x(n) = i) < \infty$, and $P(x(n + 1) = i\mid \sigma(x(k): k \leq n)) = P(x(n + 1) = i|x(n))$ on $\{x(n) \in S\}$. A Markov measure $P$ is called a two-sided Markov chain if there are substochastic matrices on $S, p$ and $q$, so that $\forall i, j \in S, n \in Z, P(x(n) = i, x(n + 1) = j) = P(x(n) = i)p(i, j) = P(x(n + 1) = j)q(j, i)$. It is shown that if $p$ is a certain kind of irreducible matrix then there exists an integer $d \geq 1$ and a real $\rho > 0$ so that $P \circ \theta^{-1}_{nd} = \rho^nP \forall n \in Z$. Of particular interest is the generalization of the case $d = 1$. A two-sided Markov chain is called quasistationary if there exists $\rho > 0$ with $P \circ \theta^{-1}_n = \rho^n P$. It is shown that if $\pi$ is a measure on $S$ and $\rho > 0$, and $p$ is a substochastic matrix on $S$, with $\pi p \leq \rho \pi$, then there exists a quasistationary chain with $p$ as forward transition, $P(x(0) = i) = \pi(i),$ and $P \circ \theta^{-1}_n = \rho^n P. P$ is used to prove the Riesz decomposition theorem for $\pi$. Finally, it is shown under certain verifiable assumptions, that a quasistationary chain, restricted to $\{\alpha \leq 0\}$, is an extended chain, with transition $p$, which is a certain kind of approximate $p$-chain in the sense of Hunt.

The paper investigates the reverse time differentiation of a stochastic exponential that occurs in smoothing, when the signal is a finite state Markov process and the observation process is a diffusion.

A Malliavin calculus is developed whose scope includes point processes, pure jump Markov processes, and purely discontinuous martingales. An integration by parts formula for functionals of Poisson point processes is proved. This is used to develop a criterion for pure jump Markov processes to have a density in $L^p$. The integration by parts formula is then used to give conditions for a purely discontinuous martingale to have a jointly continuous local time $L^x_t$ that is an occupation time density with respect to Lebesgue measure.

Let $W(t)$ be a standard Wiener process with local time $L(x, t)$. Put $L(t) = L(0, t)$ and $L^\ast(t) = \sup_{-\infty < x < \infty} L(x, t)$. We study the almost sure behaviour of small increments of $L(t)$ and also, the joint behaviour of $L(t)$ and the last excursion, $U(t)$. The increment problem of $L(x, t)$ are also studied uniformly in $x$. This implies a $\lim \inf$-type law of the iterated logarithm for $L^\ast(t)$ due to Kesten (1965), in which case the exact constant, not known before, is also determined.

Let $X_1, X_2,\cdots$ be an i.i.d. sequence with $EX_1 = \mu > 0, \operatorname{var}(X_1) = \sigma^2 > 0, E \exp(sX_1) < \infty, |s| < s_1$, and partial sums $S_0 = 0, S_n = X_1 + \cdots + X_n$. For $t \geq 0$, put $N(t) = \max \{n \geq 0: S_0,\ldots, S_n \leq t\}$, i.e., $L(t) = N(t) + 1$ denotes the first-passage time of the random walk $\{S_n\}$. Starting from some analogous results for the partial sum sequence, this paper studies the almost sure limiting behaviour of $\sup_{0 \leq t \leq T - K_T} (N(t + K_T) - N(t))$ as $T \rightarrow \infty$, under various conditions on the real function $K_T$. Improvements of the Erdos-Renyi strong law for renewal processes (resp. first-passage times) are obtained as well as strong invariance principle type versions. An indefinite range between strong invariance and strong noninvariance is also treated.

Let $\{S_n\}$ be a random walk with underlying distribution function $F(x)$ and $\{\gamma_n\}$ be a sequence of constants such that $\gamma_n/n$ is nondecreasing. A universal integral test is given which determines the lower limit of $S_n/\gamma_n$ up to a constant scale for $\lim \sup \gamma_{2n}/\gamma_n < \infty$. The generalized LIL is obtained which contains the main result of Fristedt-Pruitt (1971). The rapidly growing random walks and the limit points of $\{S_n/\gamma_n\}$ are also studied.

Given a class $\mathscr{A}$ of subsets of $\lbrack 0, 1\rbrack^d$ and an array $\{X_j: \mathbf{j} \in \mathbb{Z}^d_+\}$ of independent identically distributed random variables with $EX_j = 0, EX^2_j = 1$, the (unsmoothed) partial-sum process $S_n$ is given by $S_n(A) := n^{-d/2}\sum_{j \in n A}X_j, A \in \mathscr{A}$. If for the metric $\rho(A, B) = |A \Delta B|$ the metric entropy with inclusion $N_1(\varepsilon, \mathscr{A}, \rho)$ satisfies $\int^1_0(\varepsilon^{-1} \log N_I(\varepsilon, \mathscr{A}, \rho))^{1/2} d\varepsilon < \infty$, then an appropriately smoothed version of the partial-sum process converges weakly to the Brownian process indexed by $\mathscr{A}$. This improves on previous results of Pyke (1983) and of Bass and Pyke (1984) which require stronger conditions on the moments of $X_j$.

Sample path behavior is studied for Gaussian processes $W_p$ indexed by classes $\mathscr{L}$ of subsets of a probability space $(X, \mathscr{A}, P)$ with covariance $EW_P(A)W_P(B) = P(A \cap B)$. A function $\psi$ is found in some cases such that $\lim \sup_{t\rightarrow 0}\sup\{|W_P(C)|/\psi(P(C)): C \in \mathscr{L}, P(C) \leq t\} = 1$ a.s. This unifies and generalizes the LIL and Levy's Holder condition for Brownian motion, and some results of Orey and Pruitt for the Brownian sheet.

This paper studies extreme values in infinite moving average processes $X_t = \sum_\lambda c_{\lambda - t} Z_\lambda$ defined from an i.i.d. noise sequence $\{Z_\lambda\}$. In particular this includes the ARMA-processes often used in time series analysis. A fairly complete extremal theory is developed for the cases when the d.f. of the $Z_\lambda$'s has a smooth tail which decreases approximately as $\exp\{- z^p\}$ as $z \rightarrow \infty$, for $0 < p < \infty$, or as a power of $z$. The influence of the averaging on extreme values depends on $p$ and the $c_\lambda$'s in a rather intricate way. For $p = 2$, which includes normal sequences, the correlation function $r_t = \sum_\lambda c_{\lambda - t}c_\lambda/\sum_\lambda c^2_\lambda$ determines extremal behavior while, perhaps more surprisingly, for $p \neq 2$ correlations have little bearing on extremes. Further, the sample paths of $\{X_t\}$ near extreme values asymptotically assume a specific nonrandom form, which again depends on $p$ and $\{c_\lambda\}$ in an interesting way. One use of this latter result is as an informal quantitative check of a fitted moving average (or ARMA) model, by comparing the sample path behavior predicted by the model with the observed sample paths.

It is shown that in the sense of Baire category, almost all stochastic differential equations with uniformly bounded measurable coefficients and uniformly nondegenerate diffusions have unique strong solutions.

The aim of this paper is twofold: first, to show that Poisson approximation problems for independent summands can in a natural way be treated in a suitable operator semigroup framework, allowing at the same time for an asymptotically precise evaluation of the leading term with respect to the total variation distance; second, to determine asymptotically those Poisson distributions which minimize this distance for given Bernoulli summands. Besides semigroup methods, coupling techniques as well as direct computations are used.

In this paper we extend recent results of Szekely and others on the weak limits of elementary symmetric polynomials $S^{(k_n)}_n(X_1,\cdots, X_n)$ in the case where the order $k_n$ of the polynomials is proportional to the number of variables $n$.

Let $\mathbf{x} = (x_1,\cdots, x_n)'$ be a random vector in $R^n$. Two characterizations of normality are given. One involves the existence of two linear combinations of the $\{x_j\}$ that are independent in every coordinate system. The other, which is actually a consequence of the first, assumes that $\mathbf{x}$ obeys a linear model with spherical errors and involves sufficiency of the least-squares estimator.

There are many known asymptotic estimates of the expected number of zeros of a polynomial of degree $n$ with independent random coefficients, for $n \rightarrow \infty$. The present paper provides an estimate of the expected number of times that such a polynomial assumes the real value $K$, where $K$ is not necessarily zero. The coefficients are assumed to be normally distributed. It is shown that the results are valid even for $K \rightarrow \infty$, as long as $K = O(\sqrt n)$.

Interest in lower probability has largely focussed on lower envelopes and, more particularly, on belief functions. We consider those lower probabilities that do not admit of a dominating probability measure and hence are not lower envelopes. A simple and useful family of such undominated lower probabilities is constructed. We briefly explore the geometry of several important classes of lower probabilities and note that the class of undominated lower probabilities has the dimension of the set of all lower probabilities when these are modeled as vectors. While joint experiments can always be formed from given individual experiments characterized by probability measures, the existence of joint experiments is an open question as regards characterizations by lower probabilities. We constructively show the existence of joint experiments for a wide, but not exhaustive, range of characterizations of the marginal experiments. We also consider extensions of lower probabilities and show that a lower probability (including a measure) on a finite algebra can always be extended to an undominated lower probability on a larger, but still finite algebra. Finally we construct continuous undominated extensions of lower probabilities given on finite algebras.

It is shown that the Hausdorff-metric of $\sigma$-fields--which plays an important role for uniform martingale theorems--has a surprising "additivity" property. For example this property can be used to obtain a sharpened version of a uniform inequality for conditional expectations.