Necessary and sufficient conditions are obtained for the almost sure joint continuity of the local time of a strongly symmetric standard Markov process $X$. Necessary and sufficient conditions are also obtained for the almost sure global boundedness and unboundedness of the local time and for the almost sure continuity, boundedness and unboundedness of the local time in the neighborhood of a point in the state space. The conditions are given in terms of the 1-potential density of $X$. The proofs rely on an isomorphism theorem of Dynkin which relates the local times of Markov processes related to $X$ to a mean zero Gaussian process with covariance equal to the 1-potential density of $X$. By showing the equivalence of sample path properties of Gaussian processes with the related local times, known necessary and sufficient conditions for various sample path properties of Gaussian processes are carried over to the local times. The results are used to obtain examples of local times with interesting sample path behavior.

Let $\{L^x_t, (t, x) \in R^+ \times R\}$ be the local time of a real-valued symmetric stable process of order $1 < \beta \leq 2$ and let $\{\pi(n)\}$ be a sequence of partitions of $\lbrack 0, a\rbrack$. Results are obtained for $\lim_{n\rightarrow\infty} \sum_{x_i\in\pi(n)} |L^{x_i}_t - L^{x_{i-1}}_t|^{2/(\beta-1)}$ both almost surely and in $L^r$ for all $r > 0$. Results are also obtained for a similar expression but where the supremum of the sum is taken over all partitions of $\lbrack 0, a\rbrack$ and a function other than a power is applied to the increments of the local times. The proofs use a lemma of the authors' which is a consequence of an isomorphism theorem of Dynkin and which relates sample path behavior of local times with those of associated Gaussian processes. The major effort in this paper consists of obtaining results on the $p$-variation of the associated Gaussian processes. These results are of independent interest since the associated processes include fractional Brownian motion.

In this paper we consider the multivariate equation $X_{n+1} = A_{n+1}X_n + B_{n+1}$ with i.i.d. coefficients which have only a logarithmic moment. We give a necessary and sufficient condition for existence of a strictly stationary solution independent of the future. As an application we characterize the multivariate ARMA equations with general noise which have such a solution.

We obtain criteria for the SLLN to hold for processes which are Fourier transforms of random measures. With this spectral approach, we also give criteria for the pointwise ergodic theorem to hold, for some classes of operators between $L^\alpha$-spaces, $1 \leq \alpha < + \infty$. These results apply in particular to contractions on $L^2$. Some random fields extensions are also studied.

The GI/G/1 queueing model is regarded as a functional which maps the service and interarrival time distributions onto output quantities of interest, such as the stationary waiting time distribution. For the case where the input distributions have densities, techniques from infinite-dimensional analysis are used to obtain derivatives and Taylor series expansions for the functionals. These yield approximations to the output distributions which can be viewed as nonparametric alternatives to parametric approximations such as those provided by infinitesimal perturbation analysis or the phase method.

It is now classical that the sample mean $\bar{Y}$ is known to be asymptotically normal with $\sqrt n$ norming if and only if $0 < \operatorname{Var}\lbrack Y\rbrack < \infty$ and with arbitrary norming if and only if the df of $Y$ is in the domain of attraction of the normal df. Now let $T_n = n^{-1}\sum c_{ni}h(X_{n:i})$ for order statistics $X_{n:i}$ from a $\operatorname{df} F$ denote a general $L$-statistic subject to a bit of regularity; the key condition introduced into this problem in this paper is the regular variation of the score function $J$ defining the $c_{ni}$'s. We now define a rv $Y$ by $Y = K(\xi)$, where $\xi$ is uniform (0, 1) and where $dK = J dh(F^{-1})$. Then $T_n$ is shown to be asymptotically normal with $\sqrt n$ norming if and only if $0 < \operatorname{Var}\lbrack Y\rbrack < \infty$ and with arbitrary norming if and only if the df of $Y$ is in the domain of attraction of the normal df. As it completely parallels the classical theorem, this theorem gives the right conclusion for $L$-statistics. In order to establish the necessity above, we also obtain a nice necessary and sufficient condition for the stochastic compactness of $T_n$ and give a representation formula for all possible subsequential limit laws.

Let $X_i$ be independent and identically distributed random variables with distribution $F$. Let $M^{(n)}_n \leq \cdots \leq M^{(1)}_n$ be the sample $X_1, X_2,\ldots, X_n$ arranged in increasing order, with a convention for the breaking of ties, and let $X^{(n)}_n,\ldots, X^{(1)}_n$ be the sample arranged in increasing order of modulus, again with a convention to break ties. Let $S_n = X_1 + \cdots + X_n$ be the sample sum. We consider sums trimmed by large values, $^{(r)}S_n = S_n - M^{(1)}_n - \cdots - M^{(r)}_n, r = 1,2,\ldots, n, ^{(0)}S_n = S_n,$ and sums trimmed by values large in modulus, $^{(r)}\tilde{S}_n = S_n - X^{(1)}_n - \cdots - X^{(r)}_n, r = 1,2,\ldots, n, ^{(0)}\tilde{S}_n = S_n.$ In this paper we give necessary and sufficient conditions for $^{(r)}\tilde{S}_n/|X^{(r)}_n| \rightarrow \infty$ and $^{(r)}S_n/M^{(r)}_n \rightarrow \infty$ to hold almost surely or in probability, when $r = 1,2,\ldots$. These express the dominance of the sum over the large values in the sample in various ways and are of interest in relation to the law of large numbers and to central limit behavior. Our conditions are related to the relative stability almost surely or in probability of the trimmed sum and, hence, to analytic conditions on the tail of the distribution of $X_i$ which give relative stability.

The aim of this paper is to extend Stein's method to a compound Poisson distribution setting. The compound Poisson distributions of concern here are those of the form POIS$(\nu)$, where $\nu$ is a finite positive measure on $(0, \infty)$. A number of results related to these distributions are established. These in turn are used in a number of examples to give bounds for the error in the compound Poisson approximation to the distribution of a sum of random variables.

Let $X_{n1},\ldots, X_{nn}, n \geq 1$, be independent random variables with $P(X_{ni} = 1) = 1 - P(X_{ni} = 0) = p_{ni}$ such that $\max\{p_{ni}: 1 \leq i \leq n\}\rightarrow 0$ as $n\rightarrow\infty$. Let $W_n = \sum_{1\leq k\leq n} X_{nk}$ and let $Z$ be a Poisson random variable with mean $\lambda = EW_n$. Poisson approximation for the distribution of $W_n$ dates back to 1960, when Le Cam obtained upper bounds for the total variation distance $d(W_n, Z) = \sum_{k\geq 0}|P(W_n = k) - P(Z = k)|$. Barbour and Hall (in 1984) and Deheuvels and Pfeifer (in 1986) investigated the asymptotic behavior of $d(W_n, Z)$ as $n\rightarrow\infty$ for small, moderate and large $\lambda$. Their results imply that the orders of the bounds obtained by Le Cam are best possible. Chen proved (in 1974 and 1975) that the more general variation distance $d(h, W_n, Z) = \sum_{k\geq 0}h(k)|P(W_n = k) - P(Z = k)|, h \geq 0$, converges to 0 as $n\rightarrow\infty$, provided $\lambda$ remains bounded and $Eh(Z) < \infty$. We investigate the asymptotic behavior of: (i) $Eh(W_n) - Eh(Z_\lambda)$ for real $h$; and (ii) $d(h, W_n, Z)$ for $h \geq 0$ as $n\rightarrow\infty$ for small and moderate $\lambda$, thus generalizing the corresponding results of Barbour and Hall and of Deheuvels and Pfeifer. Our method also yields a large deviation result and holds promise for successful application in the case when $X_{n1},\ldots, X_{nn}$ are dependent.

In this paper we introduce a fairly general decoupling inequality for $U$-statistics. Let $\{X_i\}$ be a sequence of independent random variables in a measurable space $(S, \mathscr{J})$, and let $\{\tilde{X}_i\}$ be an independent copy of $\{X_i\}$. Let $\Phi(x)$ be any convex increasing function for $x \geq 0$. Let $\Pi_{ij}$ be families of functions of two variables taking $(S \times S)$ into a Banach space $(D, \|\cdot\|)$. If the $f_{ij} \in \Pi_{ij}$ are Bochner integrable and $\max_{1\leq i\neq j\leq n} E\Phi\big(\sup_{f_{ij}\in\Pi_{ij}}\|f_{ij}(X_i, X_j)\|\big) < \infty,$ then, under measurability conditions, $E\Phi\big(\sup_{\mathbf{f}\in\mathbf{\Pi}}\big\|\sum_{1\leq i\neq j\leq n} f_{ij}(X_i, X_j)\big\|\big) \leq E\Phi\big(8 \sup_{\mathbf{f}\in\mathbf{\Pi}}\big\|\sum_{1\leq i\neq j\leq n} f_{ij}(X_i, \tilde{X}_j)\big\|\big),$ where $\mathbf{f} = (f_{ij}, 1 \leq i \neq j \leq n)$ and $\mathbf{\Pi} = (\Pi_{ij}, 1 \leq i \neq j \leq n)$. In the case where $\mathbf{\Pi}$ is a family of functions of two variables satisfying $f_{ij} = f_{ji}$ and $f_{ij}(X_i, X_j) = f_{ij}(X_j, X_i)$, the reverse inequality holds (with a different constant). As a corollary, we extend Khintchine's inequality for quadratic forms to the case of degenerate $U$-statistics. A new maximal inequality for degenerate $U$-statistics is also obtained. The multivariate extension is provided.

The main result of this article is a generalization of the generalized Holder inequality for functions or random variables defined on lower-dimensional subspaces of $n$-dimensional product spaces. It will be seen that various other inequalities are included in this approach. For example, it allows the calculation of upper bounds for the product measure of $n$-dimensional sets with the help of product measures of lower-dimensional marginal sets. Furthermore, it yields an interesting inequality for various cumulative distribution functions depending on a parameter $n \in \mathbb{N}$.

Consider the family of perturbed stochastic differential equations on $\mathbb{R}^d$, $X^\varepsilon_t = X^\varepsilon_0 + \sqrt{\varepsilon} \int^t_0\sigma(X^\varepsilon_s)\circ dW_s + \int^t_0 b(X^\varepsilon_s) ds,$ $\varepsilon > 0$, defined on the canonical space associated with the standard $k$-dimensional Wiener process $W$. We assume that $\{X^\varepsilon_0, \varepsilon > 0\}$ is a family of random vectors not necessarily adapted and that the stochastic integral is a generalized Stratonovich integral. In this paper we prove large deviations estimates for the laws of $\{X^\varepsilon_., \varepsilon > 0\}$, under some hypotheses on the family of initial conditions $\{X^\varepsilon_0, \varepsilon > 0\}$.

It is proved that if a rational mapping has $\infty$ as a fixed point in its Fatou set, then its Julia set has positive capacity and the equilibrium measure is invariant. If $\infty$ is attracting or superattracting, then the equilibrium measure is strongly mixing, whereas if $\infty$ is neutral, then the equilibrium measure is ergodic and has entropy zero. Lower bounds for the entropy are given in the attracting and superattracting cases. If the Julia set is totally disconnected, then the equilibrium measure is Gibbs and therefore Bernoulli. The proofs use an induced action by the rational mapping on the space of Brownian paths started at $\infty$.

Differentiability of functionals of the empirical distribution function is extended. The supremum norm is replaced by $p$-variation seminorms, which are the $p$th roots of suprema of sums of $p$th powers of absolute increments of a function over nonoverlapping intervals. Frechet derivatives often exist for such norms when they do not for the supremum norm. For $1 < q < 2$, classes of functions uniformly bounded in $q$-variation are universal and uniform Donsker classes: The central limit theorem for empirical measures holds with respect to uniform convergence over such a class, also uniformly over all probability laws on the line. The integral $\int F dG$ was defined by L. C. Young if $F$ and $G$ are of bounded $p$- and $q$-variation respectively, where $p^{-1} + q^{-1} > 1$. Thus the normalized empirical distribution function $n^{1/2}(F_n - F)$ is with high probability in sets of uniformly bounded $p$-variation for any $p > 2$, uniformly in $n$.

A class $\mathscr{F}$ of measurable functions on a probability space $(A, \mathbb{A}, P)$ is called a $P$-Donsker class and we also write $\mathscr{F} \in \operatorname{CLT}(P)$, if the empirical processes $\mathbb{X}^P_n \equiv \sqrt{n}(\mathbb{P}_n - P)$ converge weakly to a $P$-Brownian bridge $G_P$ having bounded uniformly continuous sample paths almost surely. If this convergence holds for every probability measure $P$ on $(A, \mathbb{A})$, then $\mathscr{F}$ is called a universal Donsker class and we write $\mathscr{F} \in \operatorname{CLT}(\mathbf{M})$, where $\mathbf{M} \equiv \{$all probability measures on $(A, \mathbb{A})\}$. If the convergence holds uniformly in all $P$, then $\mathscr{F}$ is called a uniform Donsker class and we write $\mathscr{F} \in \operatorname{CLT}_u(\mathbf{M})$. For many applications the latter concept is too restrictive and it is useful to focus instead on a fixed subcollection $\mathscr{P}$ of the collection $\mathbf{M}$ of all probability measures on $(A, \mathbb{A})$. If the empirical processes converge weakly to $G_P$ uniformly for all $P \in \mathscr{P}$, then we say that $\mathscr{F}$ is a $\mathscr{P}$-uniform Donsker class and write $\mathscr{F} \in \operatorname{CLT}_u(\mathscr{P})$. We give general sufficient conditions for the $\mathscr{P}$-uniform Donsker property and establish basic equivalences in the uniform (in $P \in \mathscr{P}$) central limit theorem for $\mathbf{X}_n$, including a detailed study of the equivalences to the "functional" or "process in $n$" formulations of the $\operatorname{CLT}$. We give applications of our uniform convergence results to sequences of measures $\{P_n\}$ and to bootstrap resampling methods.

We prove that the critical value for the uniform nearest particle system equals 1.

A collection of several different probabilistic processes involving trees is shown to have an unexpected unity. This makes possible a fruitful interplay of these probabilistic processes. The processes are allowed to have arbitrary parameters and the trees are allowed to be arbitrary as well. Our work has five specific aims: First, an exact correspondence between random walks and percolation on trees is proved, extending and sharpening previous work of the author. This is achieved by establishing surprisingly close inequalities between the crossing probabilities of the two processes. Second, we give an equivalent formulation of these inequalities which uses a capacity with respect to a kernel defined by the percolation. This capacitary formulation extends and sharpens work of Fan on random interval coverings. Third, we show how this formulation also applies to generalize work of Evans on random labelling of trees. Fourth, the correspondence between random walks and percolation is used to decide whether certain random walks on random trees are transient or recurrent a.s. In particular, we resolve a conjecture of Griffeath on the necessity of the Nash-Williams criterion. Fifth, for this last purpose, we establish several new basic results on branching processes in varying environments.

The contact process on an infinite homogeneous tree is shown to exhibit at least two phase transitions as the infection parameter $\lambda$ is varied. For small values of $\lambda$ a single infection eventually dies out. For larger $\lambda$ the infection lives forever with positive probability but eventually leaves any finite set. (The survival probability is a continuous function of $\lambda$, and the proof of this is much easier than it is for the contact process on $d$-dimensional integer lattices.) For still larger $\lambda$ the infection converges in distribution to a nontrivial invariant measure. For any $n$-ary tree, with $n$ large, the first of these transitions occurs when $\lambda \approx 1/n$ and the second occurs when $1/2\sqrt{n} < \lambda < e/\sqrt{n}$. Nonhomogeneous trees whose vertices have degrees varying between 1 and $n$ behave essentially as homogeneous $n$-ary trees, provided that vertices of degree $n$ are not too rare. In particular, letting $n$ go to $\infty$, Galton-Watson trees whose vertices have degree $n$ with probability that does not decrease exponentially with $n$ may have both phase transitions occur together at $\lambda = 0$. The nature of the second phase transition is not yet clear and several problems are mentioned in this regard.

We study the asymptotic shape of the occupied region for an interacting lattice system proposed recently by Diaconis and Fulton. In this model particles are repeatedly dropped at the origin of the $d$-dimensional integers. Each successive particle then performs an independent simple random walk until it "sticks" at the first site not previously occupied. Our main theorem asserts that as the cluster of stuck particles grows, its shape approaches a Euclidean ball. The proof of this result involves Green's function asymptotics, duality and large deviation bounds. We also quantify the time scale of the model, establish close connections with a continuous-time variant and pose some challenging problems concerning more detailed aspects of the dynamics.

We consider a continuous time branching random walk on the finite set $\{1,2,\ldots, N\}$ with totally symmetric diffusion jumps and some site-dependent i.i.d. random birth rates which are unbounded. We study this process as the time $t$ and the space size $N$ tend to infinity simultaneously. In the classical law of large numbers setup for spatial branching models, the growth of the population obeys an exponential limit law due to the localization of the overwhelming portion of particles in the record point of the medium. This phenomenon is analyzed further: The historical path (in space) of a typical particle picked at time $t$ (selection) is of a rather simple and special nature and becomes in the limit singular (in distribution) to the path of the underlying mean field random walk. In general, the properties of the typical path depend on the relation in which $t$ and $N$ tend to infinity.