For certain classes of Markov systems (that is, stochastic systems which have Markov representations with transition and cotransition probabilities) considered by the author in previous papers, a correspondence was established between additive functionals of any such system and measures on a certain measurable space. We now prove analogous results for arbitrary Markov systems. Measures corresponding to the additive functionals are defined on a certain $\sigma$-algebra in the product space $R \times \Omega$ where $R$ is the real line and $\Omega$ is the sample space (we call it the central $\sigma$-algebra). The theory is applicable not only to traditional processes but also to a number of generalized stochastic processes introduced by Gelfand and Ito. A situation where the observations are performed over a random time interval and the measure $P$ can be infinite is considered in the concluding section. These generalizations are of special importance for the homogeneous case which will be treated in another publication.

We discuss the uniqueness of a class of infinite particle systems known as proximity processes, with the aid of certain "dual" Markov chains. By checking whether the dual "explodes," i.e., attempts infinitely many jumps in a finite time, and how it explodes when it does, it is possible in many cases to determine whether or not there is more than one particle system with given flip rates. We then use duality to find an example of a system which is uniquely determined by its flip rates, but whose generator is not the closure of the naive operator formed from these flip rates.

In this paper we prove a theorem on the convergence rate in the Marcinkiewicz-Zygmund strong law for stationary mixing sequences. Our result gives the $r$-quick strong law and the finiteness of moments of the largest excess of boundary crossings for such sequences.

For each sequence of independent and identically distributed $\mathbb{R}^k$-valued random variables, $k \geqq 3$, with distribution $\mu$ defined on some probability space $(\Omega, \mathscr{F}, \mathbb{P})$, let $$D_n(\omega, \mu) \equiv \sup_C |\mu_n^\omega(C) - \mu(C)|,\quad n \in \mathbb{N}, \omega \in \Omega,$$ be the so-called isotrope discrepancy (at stage $n$), where $\mu_n^\omega$ denotes the $n$th empirical distribution pertaining to $\omega$ and where the supremum is taken over the class of all convex measurable sets $C \subset \mathbb{R}^k$. It is proved that almost everywhere and in the mean $D_n(\bullet)$ converges to zero as $n^{-2/(k+1)}$ (up to a logarithmic factor), provided $\mu$ is absolutely continuous with a bounded density function of compact support.

Let $X(t)$ be a stationary Gaussian process with continuous sample paths, mean zero, and a covariance function satisfying (a) $r(t) \sim 1 - C|t|^\alpha$ as $t \rightarrow 0, 0 < \alpha \leqq 2$ and $C > 0$; and (b) $r(t) \log t = o(1)$ as $t \rightarrow \infty$. Let $\{t_n\}$ be any sequence of times with $t_n \uparrow \infty$. Then, for any nondecreasing function $f$, one obtains $P\{X(t_n) > f(t_n) \mathrm{i.o.}\} = 0$ or 1 according to a certain integral test. This result both combines and generalizes the law of iterated logarithm results for discrete and continuous time processes. In particular, it is shown that any sequence $t_n$ satisfying $\lim \sup_{n\rightarrow\infty} (t_n - t_{n-1})(\log n)^{1/\alpha} < \infty$ captures continuous time in the sense that the upper and lower class functions for the law of the iterated logarithm of $X(t_n)$ are exactly the same as those for the continuous time $X(t)$. Analogous results are obtained for Brownian motion.

It is shown that the exact measure function $\Psi(h)$ of a stationary Gaussian process with spectral density function $f(\lambda)$ proportional to $(\lambda^2 + a^2)^{-(\alpha+\frac{1}{2})}, 0 < \alpha < \frac{1}{2}$, is given by $\Psi(h) = h^{1-\alpha}(\log |\log h|)^\alpha$.

The quadratic variation of functionals $F(t)$ of $n$-dimensional Brownian motion is investigated. Let $\Pi_n = \{t_1^n, t_2^n, \cdots, t^n_{l(n)}\}$ with $a = t_1^n < t_2^n < \cdots < t^n_{l(n)} = b$ be a family of partitions of the interval $\lbrack a, b\rbrack$. The limiting behavior of $Q^2(F, \Pi_n) = \sum^{l(n)-1}_{k=1} (F(t^n_{k+1}) - F(t_k^n))^2$ as $n \rightarrow \infty$, assuming $\|\Pi_n\| \rightarrow 0$, is studied. And the existence of this limit is obtained for a fairly general class of functionals of Brownian motion.

For a Wiener process with a two-dimensional parameter $\{W_z, z \in R_+^2\}$, four types of stochastic integrals: $\int \phi dW, \int \psi dW dW, \int \psi dW dz, \int \psi dz dW$, have been defined under the condition $$E \int \phi^2 dz < \infty \quad \text{and} \quad E \int \psi^2 dz dz' < \infty.$$ The main purpose of this note is to extend the definition of these stochastic integrals by replacing $E(\bullet) < \infty$ with $(\bullet) < \infty$ a.s. in these conditions. Our results are in fact even more general, allowing $W$ to be replaced by a strong martingale with appropriate properties.

R. Pyke raised the question of the convergence of series indexed by filtering sets. This paper contains a generalization of an inequality of Marcinkiewicz-Zygmund for a certain class of filtering sets, which gives rise to the theory of series for this type of set.

Recent results have estimated the error when sums of dependent nonnegative integer-valued random variables are approximated in distribution by a Poisson variable. Two problems are considered where these results can be used to provide simple solutions. The first problem studies the asymptotic behavior, as $\alpha \rightarrow 0$, of the number of independent random arcs of length $\alpha$ needed to cover a circle of unit circumference at least $m$ times $(m \geqq 1)$. The second problem deals with urn schemes.

Ergodic theorems are proved for the one dimensional translation invariant simple exclusion process, in cases in which the initial distribution is highly nontranslation invariant.

The well-known fact that planar Brownian motion does not hit points is proved anew by a stopping-time argument.

We show that every discrete-time martingale can be interpolated to give a continuous, continuous-time martingale, and provide a necessary and sufficient condition for the existence of an interpolated martingale with no flat spots.

Let $p_t, t \geqq 0$ be the probability transition semigroup for a continuous one-dimensional diffusion. We examine continuous Markov processes $\xi_s$, defined for all $-\infty < s < \infty$, which are governed by $p_t$. We determine necessary and sufficient conditions for the set of such processes governed by $p_t$ to be nontrivial, and give an example where these conditions are satisfied.

Karlin and McGregor calculated the coincidence probabilities for $n$ particles independently executing a Markov process of a certain class. This note extends their result by allowing the particles to have different stopping times. Applied to a one-dimensional clustering problem, this gives a new solution computationally simpler than previous ones.

This paper simplifies and extends previous results on the existence of an equilibrium or stationary measure for the age process associated with a semi-Markov chain: $$(\mathbf{I}_{(t)}, \mathbf{Z}_{(t)}) = \text{(last state entered before time t}$$, $$\text{duration of this last sojourn up to} t$$).

For martingales $f \in L_p (2 \leqq p < \infty)$ the inequality $\|Mf\|_p \leqq (p + 1)\|Sf\|_p$ is proved, where $Mf = \sup_n |f_n|$ is the maximal function and $S^2 = \sum_n |f_n - f_{n-1}|^2$ the martingale square function. For integer $p$ the estimate becomes $\|Mf\|_p \leqq p\|Sf\|_p$.

Let $\{F_m\}$ be a sequence of distribution functions and $X(r, n, m)$ be the $r$th order statistic in a sample of size $n$ from $F_m$. In this note we establish a relationship between the convergences of $F_m$ and $EX(r, n, m)$ as $m \rightarrow \infty$.