Let $Z$ be a countable set, $\Xi$ the set of subsets of $Z$. A $\Xi$-valued Markov process $\{\xi_t\}$ with transition function $P(t, \xi, \Gamma)$ is called additive if there exists a family $\{\xi^A_t, t \geqq 0, A \in \Xi\}$ such that for each $A, \{\xi^A_t\}$ is Markov with transition function $P$ and $\xi^A_0 = A$, and such that $\xi^{A \cup B}_t = \xi^A_t \cup \xi^B_t, A, B \in \Xi, t \geqq 0$. Additive processes include symmetric simple exclusion, voter models and all contact processes having associates. The structure of such processes is studied, their construction from sets of independent Poisson flows, and their representations by random graphs. Applications for the case $Z = Z_d$, the $d$-dimensional integers, include individual ergodic theorems for certain cases as well as lower bounds for growth rates, and some results about different kinds of criticality when $d = 1$.

Certain Markov processes on the state space of subsets of the integers have $\varnothing$ as a trap, but have an equilibrium $\nu \neq \delta_\varnothing$. In this paper we prove weak convergence to a mixture of $\delta_\varnothing$ and $\nu$ from any initial state for some of these processes. In particular, we prove that the basic symmetric one-dimensional contact process of Harris has only $\delta_\varnothing$ and $\nu$ as extreme equilibria when the infection rate is large enough in comparison to the recovery rate.

We consider several conjectures of Hammersley and Welsh in the theory of first passage percolation on the two-dimensional rectangular lattice. Our results include: (i) a proof that the time constant is zero when the atom at zero of the underlying distribution is one-half or larger; (ii) almost sure existence of routes for the unrestricted first passage times; (iii) almost sure limit theorems for the first passages $s_{0n}$ and $b_{0n}$, the reach processes $y_t$ and $y^u_t$, and the route length processes $N^s_n$ and $N^b_n$; (iv) bounds on the expected maximum height of routes for $s_{0n}$ and $t_{0n}$ when the atom at zero of the underlying distribution is one-half or larger.

Erdos and Ney introduced a discrete time annihilating particle model on the integer lattice and conjectured that, starting from an initial state of a particle at each lattice site except the origin, the probability a particle ever hits the origin is 1. This paper proves this conjecture for the continuous time version of their model.

We show that for positive recurrent Markov chains on a general state space, a geometric rate of convergence to the stationary distribution $\pi$ in a "small" region ensures the existence of a uniform rate $\rho < 1$ such that for $\pi-\mathrm{a.a.} x, \|P^n(x, \bullet) - \pi(\bullet)\| = O(\rho^n)$. In particular, if there is a point $\alpha$ in the space with $\pi(\alpha) > 0$, the result holds if $|P^n(\alpha, \alpha) - \pi(\alpha)| = O(\rho^n_\alpha)$ for some $\rho_\alpha < 1$. This extends and strengthens the known results on a countable state space. Our results are put in the more general $R$-theoretic context, and the methods we use enable us to establish the existence of limits for sequences $\{R^nP^n(x, A)\}$, as well as exhibiting the solidarity of a geometric rate of convergence for such sequences. We conclude by applying our results to random walk on a half-line.

Let $\{X_n, n \geqq 0\}$ be a Gaussian sequence with $EX_i \equiv 0; V(X_i) \equiv 1$ and $EX_iX_j = r_{ij}$. Define $M_n = \max_{0\leqq i \leqq n}X_i$ and $m_{n,k} = \max (X_i; i \in G_n)$ where $G_n = (t_1, \cdots, t_{n'})$ is a subset of $(0, 1, 2, \cdots, n)$ and $n' = \lbrack n/k\rbrack$ for some integer $k \geqq 1, \lbrack x\rbrack$ being the integral part of $x$. We show that $P\{c_n(M_n - m_{n, k}) \leqq x\} \rightarrow (1 + (k - 1)e^{-x})^{-1}$ as $n \rightarrow \infty$ for all $x \geqq 0$ where $c_n = (2 \log n)^\frac{1}{2}$, if the sequence is "moderately dependent," namely if \begin{equation*}\tag{1}(i) \sup_{ij} |r_{ij}| < \delta < 1\end{equation*} $$(ii) |r_{ij}| \leqq \rho(i - j) \text{for} |i - j| > N_0 \text{such that} \rho_n\log n = o(1).$$ Somewhat surprisingly the same result holds even though the sequence is "strongly dependent," namely if \begin{equation*}\tag{2}(i) r_{ij} = r(i - j); r_n \text{convex for} n \geqq 0 \text{and} r_n = o(1)\end{equation*} $$(ii) (r(n) \log n)^{-1} \text{is monotone and} o(1).$$

The connection between the heat equation and Brownian motion is generalized to a process related to the equation $\partial u/\partial t = (-1)^{n + 1} \partial^{2n}u/\partial x^{2n}, n \geqq 2$. The associated measure is of unbounded variation and signed; the process cannot be realized in the space of continuous functions. Stochastic integrals $\int^t_0 \varphi(x(s))(dx)^j(s), j = 1, 2, \cdots, 2n$, are defined, and an analogue of Ito's lemma for the Brownian integral is proven. Specifically, one gets $2n$ independent differentials $(dx)^j$, with $(dx)^{2n} = (-1)^{n + 1}(2n)! dt$. Applications include the derivation of the analogue of the Brownian exponential martingale $\exp\{\alpha x - \alpha^2t/2\}$ and a class of orthogonal functions which generalize the Hermite polynomials. These are followed by the Feynmann-Kac formula, distribution of the maximum function, arc-sine law, and distribution of eigenvalues. Finally, central limit theorems are proven for convergence of sums of independent random variables identically distributed by a signed measure, normalized to have first $2n - 1$ moments equal to zero and $2n$th moment equal to $(-1)^{n + 1}(2n)!$.

In this paper, we obtain a continuous parameter generalization of Doeblin's and Harris' theory of Markov processes.

Consider a set of independent identically distributed random variables indexed by $Z^d_+$, the positive integer $d$-dimensional lattice points, $d \geqq 2$. The classical Kolmogorov-Marcinkiewicz strong law of large numbers is generalized to this case. Also, convergence rates in the law of large numbers are derived, i.e., the rate of convergence to zero of, for example, the tail probabilities of the sample sums is determined.

Let $A$ be a subset of the integers and let $S_n$ be the number of heads in $n$ tosses of a $p$ coin. If $\lim_{n\rightarrow\infty} P(S_n \in A)$ exists for some $p$ then the limit exists for all $p$ and does not depend on $p$. The relation of the limit to the density of $A$ and to a similar Poisson limit is also given.

Two models for the "many-armed bandit" problem with two distributions are considered. Feldman's result is extended to these models.

Suppose $\{B_t\}_{t\geqq 0}$ is a standard 1-dimensional Brownian motion, and $f$ is a continuous function with nonaccumulating zero set. For $t \geqq 0$, let $M_t = \int^t_0 f(B_s) dB_s$. When does $M$ generate the same fields as $B$? When does $M$ generate the same fields as some Brownian motion? The answers to these questions are obtained; they involve the behavior of $f$ around its zeros. Also, either $M$ generates the same fields as some Brownian motion, or the fields of $M$ support discontinuous martingales.

Let $W_N(\mathbf{t})$ denote the $N$-parameter Brownian sheet (Wiener process) taking values in $R^1$. For $0 < T \leqq 1$, set $\Delta(T) = \{\mathbf{t} \in R^N: 0 < t_i \leqq T, i = 1, \cdots, N\}$ and let $E(x, T) = \{\mathbf{t} \in \Delta(T): W_N(\mathbf{t}) = x\}$, the set of $\mathbf{t}$ where the process is at the level $x$. Then we show that, with probability one, the Hausdorff dimension of $E(x, T)$ equals $N - \frac{1}{2}$ for all $0 < T \leqq 1$ and every $x$ in the interior of the range of $W_N(\mathbf{t}, \mathbf{t} \in \Delta(T)$. This provides an answer to a question raised earlier by Pyke.

For a sequence of independent identically distributed random variables $\{X_n\}, n = 1, 2, \cdots,$ yielding the sums $S_n = X_1 + \cdots + X_n$ let $N(x) = \sharp\{n \geqq 1: S_n \leqq x\}$. Results of Stone and the general renewal equation as treated by Feller are used to prove that under certain conditions on the common distribution function of the $X_n$'s, the variance of $N(x)$ is asymptotically like $Ax + B + o(1)$ as $x\rightarrow\infty$ for specified constants $A$ and $B$.