Let $S$ be a countable set and $p(x, y)$ be the transition probabilities for a discrete time Markov chain on $S$. Consider the motion of particles on $S$ which obey the following rules: (a) there is always at most one particle at each site in $S$, (b) particles wait independent exponential times with mean one before moving, and (c) when a particle at $x$ is to move, it moves to $X_\tau$, where $\{X_n\}$ is the Markov chain starting at $x$ with transition probabilities $p(x, y)$ and $\tau$ is the first time that $X_n = x$ or $X_n$ is an unoccupied site. This process was introduced by Spitzer, and will be called a long range exclusion process because particles may travel long distances in short times. The process is well defined for finite configurations, and we will show how to use monotonicity arguments to define it for arbitrary configurations. It is shown that the configuration in which all sites are occupied may or may not be absorbing for the process. It always is if $p(x, y)$ is translation invariant on $S = Z^d$, but if $p(x, y)$ is a birth and death chain on $S = \{0,1,2,\cdots\}$, it is absorbing if and only if $p(x, y)$ is recurrent. For each positive function $\pi(x)$ on $S$ such that $\pi P = \pi$, there is a product measure $\nu_\pi$ on $\{0, 1\}^S$ which is a natural candidate for an invariant measure for the process. When $p(x, y)$ is translation invariant on $Z^d$, it is probably the case that $\nu_\pi$ is in fact invariant if and only if $\pi$ is constant. This will be verified under a mild regularity assumption, which is automatically satisfied if $d = 1$ or 2 or if the Laplace transform of $p(o, x)$ is finite in a neighborhood of the origin.

In this paper we will study the number of particles alive at time $t$ in a one dimensional contact process $\xi^0_t$ which starts with one particle at 0 at time 0. In the case of a nearest neighbor interaction we will show that if $|\xi^0_t|$ is the number of particles and $r_t, l_t$ are the positions of the rightmost and leftmost particles (with $r_t = l_t = 0$ if $|\xi^0_t| = 0$) then there are constants $\gamma, \alpha$, and $\beta$ so that $|\xi^0_t|/t, r_t/t$, and $l_t/t$ converge in $L^1$ to $\gamma 1_\Lambda, \alpha 1_\Lambda$ and $\beta 1_\Lambda$ where $\Lambda = \{|\xi^0_t| > 0$ for all $t\}$. The constant $\gamma = \rho(\alpha - \beta)^+$ where $\rho$ is the density of the "upper invariant measure" $\xi^Z_\infty$.

A formulation of a Markov chain in a random environment is given, generalizing special cases such as branching processes, queues, birth and death chains and random walks in random environments. It is assumed that the environmental process is Markovian, each environment corresponding to a particular law of evolution on a countable state space $\mathscr{X}$. It is then shown that there is a natural three way classification of states of $\mathscr{X}$. One of the three types of states is irregular in nature, and conditions are found under which no such states exist.

Let $\{X_{ij}: i\geqslant 1, j \geqslant 1\}$ be a double sequence of i.i.d. random variables taking values in the $d$-dimensional lattice $E_d$. Also let $S_{kl} = \Sigma^k_{i=1}\Sigma^k_{i=1}\Sigma^l_{j=1}X_{ij}$. Then the range of random walk $\{S_{kl}: k \geqslant 1, l \geqslant 1\}$ up to time $(m, n)$, denoted by $R_{mn}$, is the cardinality of the set $\{S_{kl}: 1 \leqslant k \leqslant m, 1 \leqslant l \leqslant n\}$, i.e., the number of distinct points visited by the random walk up to time $(m, n)$. Let $r^{(l)}$ be the probability that the random walk never hits the origin on the time set $\{(i, l): i \geqslant 1\}$. In this paper a sufficient condition in terms of the characteristic function of $X_{11}$ is given so that $$\lim_{(m,n)\rightarrow\infty}\frac{mn - R_{mn}}{m + n} = \sum^\infty_{l=1}(1 - r^{(l)}) < \infty\quad \mathrm{a.s.}$$

Let $\{X_n: n \geqslant 1\}$ be a sequence of i.i.d. random variables with bounded continuous density or probability mass function $f(x)$. If $E(\exp(\alpha|X_1|^\beta)) < \infty$ for some $\alpha > 0$ and $0 < \beta \leqslant 1, \mu = L(X_1), c_n = o(n^{1/(2 - \beta)})$ and $h$ is a measurable function such that $M = E(|h(X_1)|\exp(\alpha|X_1|^\beta)) < \infty$, then $$E(h(X_1)|X_1 + \cdots + X_n = n\mu + c_n) = E(h(X_1)) + M\cdot O\big(\frac{1 + |c_n|}{n}\big)$$ uniformly in $h$. It follows that $$\|\mathscr{L} (X_1\mid X_1 + \cdots + X_n = n\mu + c_n) - \mathscr{L}(X_1)\|_{\operatorname{Var}} = O\big(\frac{1 + |c_n|}{n}\big).$$ Applications are given to the binomial-Poisson convergence theorem, spacings, and statistical mechanics.

For a discrete-time finite-alphabet stationary channel $\nu$ satisfying a weak continuity requirement, it is shown that there are capacities $C_s(\nu)$ and $C_b(\nu)$ which have the following operational significance. A Bernoulli source $\mu$ is transmissible over $\nu$ via sliding-block coding if and only if the entropy $H(\mu)$ of $\mu$ is no greater than $C_s(\nu); \mu$ is transmissible via block coding if and only if $H(\mu)$ is no greater than $C_b(\nu)$. The weak continuity requirement is satisfied for the $\bar{d}$-continuous channels of Gray-Ornstein as well as other channels. An example of a channel is given to show that the case $C_s(\nu) \neq C_b(\nu)$ can occur.

Numerous properties are developed of measures that are asymptotically mean stationary with respect to a possibly nonsingular and noninvertible measurable transformation on a probability space. In particular, several necessary and sufficient conditions for the measure and transformation to satisfy the ergodic theorem are given, an asymptotic form of the Radon-Nikodym theorem for asymptotically dominated measures is developed, and the asymptotic behavior of the resulting Radon-Nikodym derivatives is described. As an application we prove a Shannon-McMillan-Breiman theorem for the case considered. Several examples are given to illustrate the results.

A duality theorem of Heathcote exhibiting a relationship between first passage times of the queue length in the GI/M/1 queue and the busy period of its dual M/G/1 queue is generalized to the phase type queues GI/PH/1 and PH/G/1. The phase type distributions include a number of well-known models such as generalized Erlang and hyperexponential as special cases and form a versatile class with a number of interesting closure properties.

A strong law of large numbers for linear combinations of order statistics is proved under integrability conditions only. Together with some straightforward extensions, the theorem generalizes previous results of Wellner, Helmers and Sen.

Let $\{X_n\}$ be a sequence of i.i.d. random variables with a common symmetric distribution $F$. Let $\mathfrak{L}(Z)$ denote the distribution of a random variable $Z$, and let $\mathfrak{D}(\alpha)$ denote the domain of attraction of a stable law of characteristic exponent $\alpha$. It is assumed that $\mathfrak{L}(X^k_1) \in \mathfrak{D}(\alpha)$ for some integer $k \geqslant 2$ and $\alpha \in (0, 2].$ Let $\mathbf{S}_n$ denote the $k$-dimensional random vector whose $j$th coordinate is $\Sigma^n_{i=1} X^j_i$, and let $m = \max\{j: k\alpha/j \geqslant 2\}$. Then there exist a sequence of $k \times k$ matrices $\{A_n\}$ and a sequence of vectors $\{\mathbf{b}_n\}$ in $\mathbb{R}^k$ such that $\mathbf{A}_n\mathbf{S}_n + \mathbf{b}_n$ converges in law to a random vector $\mathbf{S}$. The first $m$ coordinates of $\mathbf{S}$ are jointly normal and are independent of the remaining $k - m$ coordinates. No pair of these remaining $k - m$ coordinates are independent, but their joint distribution is operator-stable with two orbits.