Markov chains on a countable state space are studied under the assumption that the transition probabilities $(P_n(x,y))$ constitute a stationary stochastic process. An introductory section exposing some basic results of Nawrotzki and Cogburn is followed by four sections of new results.

We commemorate W. Doeblin with a brief recollection of his work, influence and life.

Reversible nearest particle systems are certain one-dimensional interacting particle systems whose transition rates are determined by a probability density $\beta(n)$ with finite mean on the positive integers. The reversible measure for such a system is the distribution $\nu$ of the stationary renewal process corresponding to this density. In an earlier paper, we proved under certain regularity conditions that the system converges exponentially rapidly in $L_2(\nu)$ if and only if the system is supercritical. This in turn is equivalent to $\beta(n)$ having exponential tails. In this paper, we consider the critical case, and give moment conditions on $\beta(n)$ which are separately necessary and sufficient for the convergence of the process in $L_2(\nu)$ at a specified algebraic rate. In order to do so, we develop conditions on the generator of a general Markov process which correspond to algebraic $L_2$ convergence of the process. The use of these conditions is also illustrated in the context of birth and death chains on the positive integers.

We show that in one dimension, the contact process in a random environment has an "intermediate phase" in which it survives but does not grow linearly. We conjecture that this does not occur in dimensions $d > 1$.

We study the contact process, together with a version of the percolation process with one continuously varying coordinate. It is proved here that the radius of the infected cluster has an exponentially decaying tail throughout the subcritical phase. The same is true of the Lebesgue measure (in space-time) of this cluster. Certain critical-exponent inequalities are derived and the critical point of the percolation process in two dimensions is determined exactly.

An analog of conditions of Meyer and Zheng for the relative compactness (in the sense of convergence in distribution) of a sequence of stochastic processes is formulated for general separable metric spaces and the corresponding notion of convergence is characterized in terms of the convergence in the Skorohod topology of time changes of the original processes. In addition, convergence in distribution under the topology of convergence in measure is discussed and results of Jacod, Memin and Metivier on convergence under the Skorohod topology are extended.

Assuming that $\{(X_n,Y_n)\}$ is a sequence of cadlag processes converging in distribution to $(X,Y)$ in the Skorohod topology, conditions are given under which the sequence $\{\int X_n dY_n\}$ converges in distribution to $\int X dY$. Examples of applications are given drawn from statistics and filtering theory. In particular, assuming that $(U_n,Y_n) \Rightarrow (U,Y)$ and that $F_n \rightarrow F$ in an appropriate sense, conditions are given under which solutions of a sequence of stochastic differential equations $dX_n = dU_n + F_n(X_n)dY_n$ converge to a solution of $dX = dU + F(X)dY$, where $F_n$ and $F$ may depend on the past of the solution. As is well known from work of Wong and Zakai, this last conclusion fails if $Y$ is Brownian motion and the $Y_n$ are obtained by linear interpolation; however, the present theorem may be used to derive a generalization of the results of Wong and Zakai and their successors.

Consider a stochastic process consisting of the pair of a position and a velocity, in a piecewise $\mathscr{L}^1 d$-dimensional domain. In the interior of the domain the dynamics are assigned by a potential and by random changes of the velocity occurring at exponentially distributed times, according to a probability distribution which may depend on the current position and velocity. On the boundary the process reflects physically (the angle of reflection equals the angle of incidence). First it is shown that the process is well defined for all times. Then, when the coefficients depend on a diverging parameter $N$, in particular such that the speed and the jump rate of the velocity go to $\infty$ with order $\sqrt{N}$ and at least $N$ respectively, a diffusion approximation is sought. The position process is represented as a solution of a Skorohod reflection equation: A skewing effect on the boundary results from the interaction between the dynamics and the reflection law, so that the direction of reflection is in general oblique. The assumption that the mean change of the velocity in the interior is linear in the current velocity, up to order at least $1/2$ in $1/N$, ensures that the cone of directions of reflection is independent of $N$. The continuity properties of the Skorohod oblique reflection problem enable one to show tightness of the position processes without having to estimate explicitly local times (boundary layers) and, together with a suitable law of large numbers for the velocity, allow one to identify the limit stochastic differential equation with oblique reflection. The theory is illustrated by several applications, in particular one to a mechanical model of Brownian motion.

We derive conditions under which a sequence of processes will converge to a (continuous-time) Markov chain with an entrance boundary. Our main application of this result is in proving weak convergence of the so-called population ancestral processes, associated with a wide class of exchangeable reproductive models, to a particular death process with an entrance boundary at infinity. This settles a conjecture of Kingman. We also prove weak convergence of the absorption times of many neutral genetics models to that of the Wright-Fisher diffusion, and convergence of population line-of-descent processes to another death process.

In this paper, we study stochastic differential equations with boundary conditions at the endpoints of a time interval (instead of the customary initial condition). We present existence and uniqueness results and study the Markov property of the solution. In the one-dimensional case, we prove that the solution is a Markov field $\operatorname{iff}$ the drift is affine.

Multiple stochastic integrals with respect to an infinitely divisible symmetric random measure are constructed for integrands taking values in a Banach space.

We start from a model of a branching particle system with immigration and with death rate and branching mechanism depending on time and location. Then we consider a limit case when the mass of particles and their life times are small and their density is high. This way, we construct a measure-valued process $X_t$ which we call a superprocess. Replacing the underlying Markov process $\xi_t$ by the corresponding "historical process" $\xi_{\leq t}$, we construct a measure-valued process $M_t$ in functional spaces which we call a historical superprocess. The moment functions for superprocesses are evaluated. Linear positive additive functionals are studied. They are used to construct a continuous analog of a random tree obtained by stopping every particle at a time depending on its path (say, at the first exit time from a domain). A related special Markov property for superprocesses is proved which is useful for applications to certain nonlinear partial differential equations. The concluding section is devoted to a survey of the literature, and the terminology on Markov processes used in the paper is explained in the Appendix.

For a branching Brownian motion, a probability space of trees is defined. By analogy with stopping times on $\mathbb{R}$, stopping lines are defined to get a general branching property. We exhibit an intrinsic class of martingales which are products indexed by the elements of a stopping line. We prove that all these martingales have the same limit which we identify. Two particular cases arise: the line of particles living at time $t$ and the first crossings of a straight line whose equation is $y = at - x$ in the plane $(y,t)$.

In this paper we describe a general device that allows us to deduce various kinds of theorems (moment estimates, exponential inequalities, strong law of large numbers, stability results, bounded law of the iterated logarithm) for partial sums of independent vector-valued random variables from related results for partial sums of independent real-valued random variables. The concept of majorization will play a key role in our considerations.

We study the almost sure behavior of sums of $\operatorname{iid}$ random variables satisfying the bounded LIL in Hilbert space. We show that the almost sure behavior is different from the Gaussian case, whenever the second strong moments are infinite. A law of the $k$ times iterated logarithm is established which refines the bounded LIL. The interesting feature here is that contrary to the known conditions for the bounded LIL, one needs not only moment type conditions but also a nice structure of the covariance operator.

We study the behaviour of $\mu\{x \in E; \|x\| > t\}$ as $t \rightarrow \infty$ for a Gaussian measure $\mu$ in a Banach or quasi-Banach space in the following cases: 1. $E = l_p, 2 < p < \infty$, and $\mu$ of diagonal form but not necessarily symmetric; 2. $E =$ Hilbert space and $\mu$ arbitrary; 3. $E = l^n_p, 0 < p < 2$, and $\mu$ of diagonal form. While 2 solves a problem of Hweng (1980), 1 and 3 extend some results of Dobric, Marcus and Weber (1988).

In this paper we prove an upper large deviation bound for a general class of Markov processes, which includes processes with discontinuous statistics. We also specialize the results to a class of jump Markov processes that model scaled queuing systems.

We derive an asymptotic approximation of the joint distribution $\operatorname{prob}(N(u) - n \in A, S_{N(u)} - u \in B)$ as $n$ and $u \rightarrow \infty$. Here $N(u) = \min\{n; S_n > u\}$ denotes the first passage time for a random walk of the form $S_n = \sum^n_{k = 1}U_k(\xi_{k - 1},\xi_k)$, where $\xi_0,\xi_1,\ldots$ is a finite Markov chain and where $\{U_k(i,j)\}^\infty_{k = 1}$ is a sequence of independent random variables. The approximation holds for all sets $B$ and a fairly large class of sets $A$.

Let $\zeta$ be chosen at random on the surface of the $p$-sphere in $\mathbb{R}^n, 0_{p,n} := \{x \in \mathbb{R}^n: \sum^n_{i = 1}|x_i|^p = n\}$. If $p = 2$, then the first $k$ components $\zeta_1,\ldots, \zeta_k$ are, for $k$ fixed, in the limit as $n\rightarrow\infty$ independent standard normal. Considering the general case $p > 0$, the same phenomenon appears with a distribution $F_p$ in an exponential class $\mathscr{E}. F_p$ can be characterized by the distribution of quotients of sums, by conditional distributions and by a maximum entropy condition. These characterizations have some interesting stability properties. Some discrete versions of this problem and some applications to de Finetti-type theorems are discussed.

In this article we solve the minimax grid matching problem in dimensions greater than two. As a by-product, we settle a long-open problem involving the Glivenko-Cantelli convergence of empirical measures.

A natural exponential family $\mathscr{F}$ is characterized by the pair $(V,\Omega)$, called the variance function (VF), where $\Omega$ is the mean domain and $V$ is the variance of $\mathscr{F}$ expressed in terms of the mean. Any VF can be used to construct an exponential dispersion model, thus providing a potential generalized linear model. A problem of increasing interest in the literature is the following: Given an open interval $\Omega$ and a function $V$ defined on $\Omega$, is the pair $(V,\Omega)$ a VF of a natural exponential family? In this paper, we develop a complex analytic approach to this question and focus on VF's having meromorphic mean functions; that is, if $T$ is the Laplace transform of an element of the family, then $T'/T$ is extendable to a meromorphic function on $\mathbb{C}$. We derive properties of such VF's and characterize a class of VF's $(V,\Omega)$, where $V$ admits a unique analytic continuation in $\mathbb{C}$, except for isolated singularities. (Included in this class are VF's having $V$'s that admit meromorphic continuation to $\mathbb{C}$.) We show that this class equals the set of VF's which are at most second degree polynomials. We also investigate the class in which $V$ has the form $P + Q\sqrt{R}$, where $P$ and $Q$ are arbitrary rational functions and $R$ is a polynomial of at most second degree. We characterize all VF's in this class for which the mean function is meromorphic and show that $P = kR$ for some constant $k$ and $Q$ is a polynomial of at most first degree. Throughout the paper, we demonstrate the wide applicability of our results by showing that many classes of simple-form pairs $(V,\Omega)$ can be excluded from being VF's.

Given a partial order of $N$ items, a linear extension that is almost uniformly distributed, in the sense of variation distance, is generated. The algorithm runs in polynomial time. The technique used is a coupling for a random walk on a polygonal subset of the unit sphere in $\mathbb{R}^N$. Including is a discussion of how accurately the steps of the random walk must be computed.