For very general random Fourier series and infinitely divisible processes on a locally compact Abelian group $G$, a necessary and sufficient condition for sample continuity is given in terms of the convergence of a certain series. This series expresses a control on the covering numbers of a compact neighborhood of $G$ by certain nonrandom sets naturally associated with the Fourier series (resp. the process). In the nonstationary case, we give a necessary Sudakov-type condition for a probability measure in a Banach space to be a Levy measure.

In this paper we develop an asymptotic expansion for the $\varepsilon$-neighborhood of the symmetric stable process of order $\beta, 1 < \beta < 2$. Our expansion is in powers of $\varepsilon^{2-\beta}$ with the $n$th coefficient related to $n$-fold self-intersections of our stable process.

Let $\mathbf{X}(t)$ be a stationary vector Gaussian process in $R^m$ whose components are independent copies of a real stationary Gaussian process with covariance function $r(t)$. Let $\phi(z)$ be the standard normal density and, for $t > 0, \varepsilon > 0$, consider the double integral $\int^t_0\int^t_0\varepsilon^{-m} \prod^m_{j=1} \phi(\varepsilon^{-1}(X_j(s) - X_j(s')))ds ds',$ which represents an approximate self-intersection local time of $\mathbf{X}(s), 0 \leq s \leq t$. Under the sole condition $r \in L_2$, the double integral has, upon suitable normalization, a limiting normal distribution under a class of limit operations in which $t \rightarrow \infty$ and $\varepsilon = \varepsilon(t)$ tends to 0 sufficiently slowly. The expected value and standard deviation of the double integral, which are the normalizing functions, are asymptotically equal to constant multiples of $t^2$ and $t^{3/2}$, respectively. These results are valid without any restrictions on the behavior of $r(t)$ for $t \rightarrow 0$ other than continuity.

We define a Levy process on a $d$-dimensional hierarchical lattice. By construction the Green's function for this process decays as $|x|^{2-d}$. For $d = 4$, we prove that the introduction of a sufficiently weak self-avoidance interaction does not change this decay provided the mass $\equiv$ "killing" rate is chosen in a special way, so that the process is critical.

We show that the transience or recurrence of a random walk in certain random environments on an arbitrary infinite locally finite tree is determined by the branching number of the tree, which is a measure of the average number of branches per vertex. This generalizes and unifies previous work of the authors. It also shows that the point of phase transition for edge-reinforced random walk is likewise determined by the branching number of the tree. Finally, we show that the branching number determines the rate of first-passage percolation on trees, also known as the first-birth problem. Our techniques depend on quasi-Bernoulli percolation and large deviation results.

In a discrete-time supercritical branching random walk, let $Z^{(n)}$ be the point process formed by the $n$th generation. Let $m(\lambda)$ be the Laplace transform of the intensity measure of $Z^{(1)}$. Then $W^{(n)}(\lambda) = \int e^{-\lambda x}Z^{(n)}(dx)/m(\lambda)^n$, which is the Laplace transform of $Z^{(n)}$ normalized by its expected value, forms a martingale for any $\lambda$ with $|m(\lambda)|$ finite but nonzero. The convergence of these martingales uniformly in $\lambda$, for $\lambda$ lying in a suitable set, is the first main result of this paper. This will imply that, on that set, the martingale limit $W(\lambda)$ is actually an analytic function of $\lambda$. The uniform convergence results are used to obtain extensions of known results on the growth of $Z^{(n)}(nc + D)$ with $n$, for bounded intervals $D$ and fixed $c$. This forms the second part of the paper, where local large deviation results for $Z^{(n)}$ which are uniform in $c$ are considered. Finally, similar results, both on martingale convergence and uniform local large deviations, are also obtained for continuous-time models including branching Brownian motion.

In this article we propose and study finite reversible nearest particle systems in inhomogeneous and random environments. Using the Dirichlet principle and the ergodic theorem we prove that a finite reversible nearest particle system in a random environment determined by an i.i.d. sequence $\lambda_i$ survives if $E \log \lambda_i > 0$ and dies out if $E\lambda_i < 1$. Some discussion is provided to show that both survival and extinction may happen when $E \log \lambda_i < 0$ and $E \lambda_i > 1$.

We consider some deterministic cellular automata on the state space $\{0, 1\}^{\mathbb{Z}^d}$ evolving in discrete time, starting from product measures. Basic features of the dynamics include: 1's do not change, translation invariance, attractiveness and nearest neighbor interaction. The class of models which is studied generalizes the bootstrap percolation rules, in which a 0 changes to a 1 when it has at least $l$ neighbors which are 1. Our main concern is with critical phenomena occurring with these models. In particular, we define two critical points: $p_c$, the threshold of the initial density for convergence to total occupancy, and $\pi_c$, the threshold for this convergence to occur exponentially fast. We locate these critical points for all the bootstrap percolation models, showing that they are both 0 when $l \leq d$ and both 1 when $l > d$. For certain rules in which the orientation is important, we show that $0 < p_c = \pi_c < 1$, by relating these systems to oriented site percolation. Finally, these oriented models are used to obtain an estimate for a critical exponent of these models.

We apply an invariance principle due to De Masi, Ferrari, Goldstein and Wick to the edge process for critical reversible nearest-particle systems. Their result also gives an upper bound for the diffusion constant that we compute explicitly. A comparison between the movement of the edge, when the other particles are frozen, and a random walk allows us to find a lower bound for the diffusion constant. This shows that the right renormalization for the edge to converge to a nondegenerate Brownian motion is the usual one. Note that analogous results for nearest-particle systems are only known for the contact process in the supercritical case.

We obtain the decay rate of the large deviation probabilities of occupation time for the symmetric simple exclusion process. Furthermore, in dimension $d \neq 2$, we prove a large deviation principle for the occupation time. To obtain these results, we prove hydrodynamical limits for the weakly asymmetric simple exclusion process and we prove a large deviation principle for the empirical density for the symmetric simple exclusion process.

In this paper we study a growth model known as the "contact process with sexual reproduction" on $\mathbf{Z}^2$. We focus on the "symmetric" model in which a "child particle" can be produced at a vacant site whenever a pair of its neighboring sites is occupied by "parent particles". Two kinds of stability of the absorbing state $\phi$ (i.e., the state in which all the sites are vacant) are investigated in this paper. The first kind of stability concerns the behavior of the system when it starts close to the state $\phi$. More explicitly, we consider the system starting with a random configuration in which the sites are occupied independently with occupation probability $p$, where $p$ is a small positive parameter. The system is said to be stable if, for sufficiently small $p$, the probability that a site is occupied approaches 0 as time approaches infinity. The second kind of stability concerns the behavior of the system under the perturbation of adding a small quantity $\beta > 0$ to all the birth rates ("spontaneous birth at rate $\beta$"). In this case, stability means that there is an equilibrium state which is close to $\phi$ when $\beta$ is small. It is proven in this paper that in the symmetric model the state $\phi$ is stable under the first kind of perturbation, but it is unstable under the second kind of perturbation.

In this paper we use a martingale problem characterization to study the behavior of finite measure valued superprocesses with a variety of spatial motions. In general the superprocess, when normalized to be a probability, will converge to a point mass at its extinction time. For some spatial motions we prove that there are times near extinction at which the closed support of the process is concentrated near one point. We obtain a Tanaka formula for the measure of a half space under a one dimensional symmetric stable superprocess of index $\alpha$ and we show this process fails to be a semimartingale if $1 < \alpha \leq 2$.

A verifiable criterion is derived for the stability in distribution of singular diffusions, that is, for the weak convergence of the transition probability $p(t; x, dy)$, as $t \rightarrow \infty$, to a unique invariant probability. For this we establish the following: (i) tightness of $\{p(t; x, dy): t \geq 0\}$; and (ii) asymptotic flatness of the stochastic flow. When specialized to highly nonradial nonsingular diffusions the results here are often applicable where Has'minskii's well-known criterion fails. When applied to traps, a sufficient condition for stochastic stability of nonlinear diffusions is derived which supplements Has'minskii's result for linear diffusions. We also answer a question raised by L. Stettner (originally posed to him by H. J. Kushner): Is the diffusion stable in distribution if the drift is $Bx$ where $B$ is a stable matrix, and $\sigma(\cdot)$ is Lipschitzian, $\sigma(\underline{0}) \neq 0$? If not, what additional conditions must be imposed?

The population genealogical processes associated with a wide range of exchangeable reproductive models (including the Wright-Fisher model) are shown to converge weakly, as the population size becomes large, to a particularly tractable limiting process, the age-ordered analog of Kingman's coalescent. This result extends the known convergence results for sample processes and effectively completes the robustness theory for neutral genealogies. Its consequences, which include a unification of the results for neutral models, have already been exploited elsewhere. The techniques used rely heavily on knowledge of sample behavior, together with consistency arguments. They may be of more general interest.

Following Choquet, the capacity associated with a Markov process is said to be dichotomous if each compact set $K$ contains two disjoint sets with the same capacity as $K$. In the context of right processes, we prove that the dichotomy of capacity is equivalent to Hunt's hypothesis that semipolar sets are polar. We also show that a weaker form of the dichotomy is valid for any Levy process with absolutely continuous potential kernel.

This paper is concerned with a class of stochastic recursive sequences that arise in various branches of queueing theory. First, we make use of Kingman's subadditive ergodic theorem to determine the stability region of this type of sequence, or equivalently, the condition under which they converge weakly to a finite limit. Under this stability condition, we also show that these sequences admit a unique finite stationary regime and that regardless of the initial condition, the transient sequence couples in finite time with this uniquely defined stationary regime. When this stability condition is not satisfied, we show that the sequence converges a.s. to $\infty$ and that certain increments of the process form another type of stochastic recursive sequence that always admit at least one stationary regime. Finally, we give sufficient conditions for this increment sequence to couple with this stationary regime.

Stochastic Petri networks provide a general formalism for describing the dynamics of discrete event systems. The present paper focuses on a subclass of stochastic Petri networks called stochastic event graphs, under the assumption that the variables used for their "timing" form stationary and ergodic sequences of random variables. We show that such stochastic event graphs can be seen as a $(\max, +)$ linear system in a random, stationary and ergodic environment. We then analyze the associated Lyapounov exponents and construct the stationary and ergodic regime of the increments, by proving an Oseledec-type multiplicative ergodic theorem. Finally, we show how to construct the stationary marking process from these results.

Some explicit isomorphisms are constructed between Bernoulli shifts with infinite entropy.

Grassberger suggested an interesting entropy estimator, namely, $\frac{n \log n}{\sum^n_{i=1} L^n_i},$ where $L^n_i$ is the shortest prefix of $x_i, x_{i+1},\ldots$, which is not a prefix of any other $x_j, x_{j+1},\ldots,$ for $j \leq n$. We show that this estimator is not consistent for the general ergodic process, although it is consistent for Markov chains. A weaker trimmed mean type result is proved for the general case, namely, given $\varepsilon > 0$, eventually almost surely all but an $\varepsilon$ fraction of the $L^n_i/\log n$ will be within $\varepsilon$ of $1/H$. A related Hausdorff dimension conjecture is shown to be false.

A probability measure on the sphere is absolutely continuous with respect to the uniform measure on the sphere if and only if the probability of any open set varies continuously as the sphere is rotated. In general, if a topological group $G$ acts transitively on a topological space $S$, and both are Hausdorff, locally compact and second countable, then a probability measure $\nu$ on the Borel sets of $S$ is absolutely continuous with respect to the unique invariant measure class on $S$ if and only if the $\nu$-probability of an open set in $S$ varies continuously under the action of the group $G$. If $S$ is a Borel $G$-space, but the action is not assumed to be transitive, then $\nu(gE)$ is continuous in $g$ for every Borel set $E$ if and only if $\nu$ is absolutely continuous with respect to a quasi-invariant measure on $S$.

Starting with a Borel probability measure $P$ on $X$ (where $X$ is a separable Banach space or a compact metrizable convex subset of a locally convex topological vector space), the class $\mathscr{F}(P)$, called the fusions of $P$, consists of all Borel probability measures on $X$ which can be obtained from $P$ by fusing parts of the mass of $P$, that is, by collapsing parts of the mass of $P$ to their respective barycenters. The class $\mathscr{F}(P)$ is shown to be convex, and the ordering induced on the space of all Borel probability measures by $Q \prec P$ if and only if $Q \in \mathscr{F}(P)$ is shown to be transitive and to imply the convex domination ordering. If $P$ has a finite mean, then $\mathscr{F}(P)$ is uniformly integrable and $Q \prec P$ is equivalent to $Q$ convexly dominated by $P$ and hence equivalent to the pair $(Q, P)$ being martingalizable. These ideas are applied to obtain new martingale inequalities and a solution to a cost-reward problem concerning optimal fusions of a finite-dimensional distribution.

Let $\{X_j\}$ be independent, identically distributed random variables with continuous nondegenerate distribution $F$ which is symmetric about the origin. Let $\{X_n(1), X_n(2),\ldots, X_n(n)\}$ denote the arrangement of $\{X_1,\ldots, X_n\}$ in decreasing order of magnitude, so that with probability 1, $|X_n(1)| > |X_n(2)| > \cdots > |X_n(n)|$. For integers $r_n \rightarrow \infty$ such that $r_n/n \rightarrow 0$, define the self-normalized trimmed sum $T_n = \sum^n_{i=r_n}X_n(i)/\{\sum^n_{i=r_n}X^2_n(i)\}^{1/2}$. The asymptotic behavior of $T_n$ is studied. Under a probabilistically meaningful analytic condition generalizing the asymptotic normality criterion for $T_n$, various interesting nonnormal limit laws for $T_n$ are obtained and represented by means of infinite random series. In general, moreover, criteria for degenerate limits and stochastic compactness for $\{T_n\}$ are also obtained. Finally, more general results and technical difficulties are discussed.

Skorohod has shown that the convergence of sums of i.i.d. random variables to an $\alpha$-stable Levy motion, with $0 < \alpha < 2$, holds in the weak-$J_1$ sense. $J_1$ is the commonly used Skorohod topology. We show that for sums of moving averages with at least two nonzero coefficients, weak-$J_1$ convergence cannot hold because adjacent jumps of the process can coalesce in the limit; however, if the moving average coefficients are positive, then the adjacent jumps are essentially monotone and one can have weak-$M_1$ convergence. $M_1$ is weaker than $J_1$, but it is strong enough for the $\sup$ and $\inf$ functionals to be continuous.

In this paper we establish a large deviations principle for the non-Gaussian stochastic reaction-diffusion equation (SRDE) $\partial_t\nu^\varepsilon = \mathscr{L}\nu^\varepsilon + f(x, \nu^\varepsilon) + \varepsilon\sigma(x, \nu^\varepsilon)\ddot{W}_{tx}$ as a random perturbation of the deterministic RDE $\partial_t\nu^0 = \mathscr{L}\nu^0 + f(x, \nu^0)$. Here the space variable takes values on the unit circle $S^1$ and $\mathscr{L}$ is a strongly-elliptic second-order operator with constant coefficients. The functions $f$ and $\sigma$ are sufficiently regular so that there is a unique solution to the above SRDE for any continuous initial condition. We also assume that there are positive constants $m$ and $M$ such that $m \leq \sigma(x, y) \leq M$ for all $x$ in $S^1$ and all $y$ in $\mathbb{R}$. The perturbation $\ddot{W}_{tx}$ is the formal derivative of a Brownian sheet. It is known that if the initial condition is continuous, then the solution will also be continuous, and moreover, if the initial condition is assumed to be Holder continuous of exponent $\kappa$ for some $0 < \kappa < \frac{1}{2}$, then the solution will be Holder continuous of exponent $\kappa/2$ as a function of $(t, x).$ In this paper we establish the large deviations principle for $\nu^\varepsilon$ in the Holder norm of exponent $\kappa/2$ when the initial condition is Holder continuous of exponent $\kappa$ for any $0 < \kappa < \frac{1}{2}$, and when the initial condition is assumed only to be continuous, we establish the large deviations principle for $\nu^\varepsilon$ in the supremum norm. Moreover, we prove that these large deviations principles are uniform with respect to the initial condition.

Let $X$ be a Hunt process starting from a regular recurrent point 0 and $\nu$ a smooth probability measure on the state space. We show that $T = \inf\{s: A_s > L_s\}$, where $A$ is the continuous additive functional associated to $\nu$ and $L$ the local time at 0, solves the Skorokhod problem for $\nu$, that is, $X_T$ has law $\nu$. We construct another solution which minimizes $\mathbb{E}_0(B_S)$ among all the solutions $S$ of the Skorokhod problem, where $B$ is any positive continuous additive functional. The special case where $X$ is a symmetric Levy process is discussed.