Let $\{\xi_t\}$ be a Markov process whose values are subsets of $Z_d$, the $d$-dimensional integers. Put $\xi_t(x) = 1$ if $x \in \xi_t$ and 0 otherwise. The transition intensity for a change in $\xi_t(x)$ depends on $\{\xi_t(y), y$ a neighbor of $x\}$. The chief concern is with "contact processes," where $\xi_t(x)$ can change from 0 to 1 only if $\xi_t(y) = 1$ for some $y$ neighboring $x$. Let $p_t(\xi) = \operatorname{Prob} \{\xi_t \neq \varnothing \mid \xi_0 = \xi\}$. Under appropriate conditions, $p_t$ is increasing, subadditive, or submodular in $\xi$. In the case of contact processes, conditions are giving implying that $p_\infty(\xi) = 0$ for all finite $\xi$, or that the contrary is true. In other cases conditions for ergodicity are given.

Let $p(x, y)$ be the transition function for an irreducible, positive recurrent, reversible Markov chain on the countable set $S$. Let $\eta_t$ be the infinite particle system on $S$ with the simple exclusion interaction and one-particle motion determined by $p$. The principal result is that there are no nontrivial invariant measures for $\eta_t$ which concentrate on infinite configurations of particles on $S$. Furthermore, it is proved that if the system begins with an arbitrary infinite configuration, then it converges in probability to the configuration in which all sites are occupied.

Let $X(t), 0 \leqq t \leqq 1$, be a real Gaussian process with mean 0 and continuous sample functions. For $u > 0$, form the process $u(X(t) - u)$. In this paper two related problems are studied. (i) Let $G$ be a nonnegative measurable function, and put $L = \int^1_0 G(u(X(t) - u)) dt$. For certain classes of processes $X$ and functions $G$, we find, for $u \rightarrow \infty$, the limiting conditional distribution of $L$ given that it is positive. (ii) For the same class of processes $X$, we find the asymptotic form of $P(\max_{\lbrack 0,1 \rbrack} X(t) > u)$ for $u \rightarrow \infty$. Finally, these results are extended to the process with the "moving barrier," $X(t) - f(t)$, where $f$ is a continuous function.

It is shown that there is a close relationship between the convergence of a sequence of normalized Galton-Watson processes and the convergence of the rowsums of a certain triangular array of independent identically distributed random variables. Using this result some limit theorems by Jirina and Lamperti are strengthened.

Let $X(t); t \geqq 0$ be a stationary continuous-time Markov chain with state space $\{1,2,\cdots, N\}$ and jump times $t_1, t_2,\cdots$. Let $T_\alpha(t); t \geqq 0, 1 \leqq \alpha \leqq N$, be semi-groups and $\Pi_{jk} (u); u \geqq 0, 1 \leqq j \neq k \leqq N$, operators defined on Banach space $B$. Under suitable conditions on these operators, including commutativity, and an appropriate time change in $\varepsilon > 0$ on $X(t)$, we give limiting behavior for the discontinuous random evolutions $T_{X(0)}(t_1^\varepsilon) \Pi_{X(0)X(t_1)} (\varepsilon)T_{X(t_1)}(t_2^\varepsilon - t_1^\varepsilon)\cdots T_{X(t_\nu)}(t - t_\nu^\varepsilon)$ as $\varepsilon \rightarrow 0$. By considering the `expectation semi-group' of the discontinuous random evolutions, we prove a type of singular perturbation theorem and give formulas for the asymptotic solution. These results rely on a limit theorem for the joint distribution of the occupation-time and number-of-jump random variables of the chain $X(\bullet)$. We prove this theorem and with `random evolution' techniques use it to give new proofs of limit theorems for Markov processes on $N$ lines. Analogous results are obtained when the controlling process is a discrete-time finite-state Markov chain.

We consider a family $X_n^\theta$ of discrete-time Markov processes indexed by a positive "step-size" parameter $\theta$. The conditional expectations of $\Delta X_n^\theta, (\Delta X_n^\theta)^2$, and $|\Delta X_n^\theta|^3$, given $X_n^\theta$, are of the order of magnitude of $\theta, \theta^2$, and $\theta^3$, respectively. Previous work has shown that there are functions $f$ and $g$ such that $(X_n^\theta - f(n\theta))/\theta^{\frac{1}{2}}$ is asymptotically normally distributed, with mean 0 and variance $g(t)$, as $\theta \rightarrow 0$ and $n\theta \rightarrow t < \infty$. The present paper extends this result to $t = \infty$. The theory is illustrated by an application to the Wright-Fisher model for changes in gene frequency.

Boundary conditions in the form of equalities have been used by Feller, Dynkin, and others to characterize the range of the resolvent operator for certain continuous time Markov chains. Along similar lines Denzel, Kemeny, and Snell were able to establish a characterization and Riesz decomposition for the excessive functions of a more restricted class of Markov chains through the use of boundary conditions in the form of inequalities. The present paper sets out to clarify and build upon this work by reanalyzing these excessive functions in a more general setting. Here the boundary theory developed by Chung is brought to bear on the problem so that the results can be derived in canonical form for probabilistic interpretation.

We prove theorems on weak convergence of random elements $X_m(\omega, t), 0 \leqq t \leqq 1$, to a Gaussian process. In Part I, these random elements are constructed on the basis of the linear means of a lacunary trigonometric series $\sum a_j \cos n_j\omega$. In Part II, the lacunarity hypothesis is dropped and replaced by the hypothesis of linear independence of the real numbers $n_j$.

Suppose $X$ and $X_n, n \geqq 1$, are i.i.d. random variables whose common distribution lies in the domain of attraction of a completely asymmetric stable law of index $\alpha (0 < \alpha < 2)$, so that (i) as $\nu \rightarrow \infty, \nu \rightarrow P\{X \geqq \nu\}$ varies regularly with exponent $-\alpha$, and (ii) $\lim_{\nu\rightarrow\infty} P\{X \leqq - \nu\}/P\{X \geqq \nu\} = 0$. Under a condition only slightly more strigent than (ii), we present Strassen-type functional laws of the iterated logarithm for the partial sums $S_n = \sum_{m\leqq n} X_m, n \geqq 1$. Our laws hold in particular when $X \geqq 0$; the proofs in this case utilize some new large deviation results for the $S_n$'s.

Let $Z_i = (X_i, Y_i), i \geqq 1$, be independent two-dimensional random variables, defined on a probability triple $(\Omega, \mathscr{A}, P)$, such that $E(X_i) = E(Y_i) = E(X_i Y_i) = 0, E(X_i^2) < \infty, E(Y_i^2) < \infty$ for all $i$. The purpose of this paper is to investigate the limit points of $\{(S_n(\omega)/L(n), T_n(\omega)/M(n)), n = 1,2,\cdots\}$, where $\omega \in \Omega, S_n = \sum^n_{i=1} X_i, T_n = \sum^n_{i=1} Y_i, L(n) = \lbrack 2E(S_n^2) \log \log E(S_n^2) \rbrack^{\frac{1}{2}}, M(n) = \lbrack 2E(T_n^2) \log \log E(T_n^2) \rbrack^{\frac{1}{2}}$. The author will show the limit sets are the closed unit disk almost surely under some general conditions. An example with all limit points lying on the two axes with probability one will be constructed.

Robbins and Siegmund have made use of the martingale $$\int^\infty_0 \exp(yW(t) - \frac{1}{2}ty^2) dF(y), t \geqq 0$$, to evaluate the probability that the Wiener process $W(t)$ would ever cross certain boundaries which are moving with time. By making use of martingales of the form $u(X(t), t)$, we apply the Robbins-Siegmund method to find boundary crossing probabilities for other Markov processes $X(t)$. The question of when $u(X(t), t)$ is a martingale is first studied. We generalize a result of Doob based on semigroups of type $\Gamma$, and we examine in particular the situations for stochastic integrals and processes with stationary independent increments.

A generalization of the weak law of large numbers is proved in the case of a process indexed by a second countable locally compact Abelian group. An application is given to random measures.

Let $S_n = X_1 + \cdots + X_n$ where $\{X_k\}_{k \geqq 1}$ is a sequence of independent, identically distributed random variables with mean zero and variance one. By the Skorohod representation $S_n$ has the same distribution as $\chi(U_n)$ where $\chi$ is standard Brownian motion. We find increasing sequences of real numbers $\{c_n\}$ and $\{d_n\}$ such that $$\lim \sum_{n\rightarrow\infty} \frac{\chi(U_n) - \chi(n)}{c_n \operatorname{lg} n)^{\frac{1}{2}}} = \infty \text{a.s}$$ and $$\lim \sup_{n\rightarrow\infty} \frac{\chi(U_n) - \chi(n)}{(d_n \operatorname{lg} n)^{\frac{1}{2}}} = 0 \text{a.s.}$$ We conclude with an example which explicitly gives the sequences $\{c_n\}$ and $\{d_n\}$ in terms of original random variables $\{X_k\}$.

An improvement in the rate of convergence for some functionals in probability given by Sawyer when a moment of specified order exists is obtained.

It is shown that, if, for a distribution function $F, 1 - F(x) + F(-x)$ varies regularly at $\infty$ with exponent $\alpha, 0 > \alpha > -1$, then $|\operatorname{Im} \phi(t)| = O(I - \operatorname{Re} \phi(t)) (t \rightarrow 0)$, where $\phi$ is the characteristic function of $F$. Versions for $\alpha \leqq -1$ are also given.