Let $u_1(x)$ be the 1-potential kernel density for a Levy process, let $\phi^2(x) = 2u_1(0) - u_1(x) - u_1(-x)$, let $\bar{\phi}$ be the monotone rearrangement of $\phi$ and let $I(\bar{\phi}) = \int_{0+} \phi(u)u^{-1}(\log(1/u))^{-1/2} du$. Barlow and Hawkes proved that if $I(\bar{\phi}) < \infty$, then the local time has a jointly continuous version. In this paper it is shown that if $I(\bar{\phi}) < \infty$, then the local time is not continuous.

We study two processes obtained as follows: Take two independent $d$-dimensional Brownian motions started at points $x, y$, respectively. For the first process, let $d \geq 3$ and condition on $X_t = Y_t$ for some $t$ (a set of probability 0). Run $X$ out to the point of intersection and then run $Y$ in reversed time from this point back to $y$. For the second process, let $d \geq 5$ and perform the same construction, this time conditioning on $X_s = Y_t$ for some $s, t$. The first process is shown to be Doob's conditioned (to go from $x$ to $y$) Brownian motion $Z$, and the second has distribution absolutely continuous with respect to that of $Z$, the Radon-Nikodym density being a constant times the time $Z$ takes to travel from $x$ to $y$. Similar results (including extensions to the critical dimensions $d = 2$ and $d = 4$) are obtained by conditioning the motions to hit before they leave domains. We use the asymptotics of the probability of "near misses" and results on the weak convergence of $h$-transforms.

Integral tests are found for the uniform escape rate of a $d$-dimensional Brownian path $(d \geq 4)$, i.e., for the lower growth rate of $\inf\{|X(t) - X(s)|: 0 \leq s, t \leq 1, |t - s| \geq h\}$ as $h \downarrow 0$. The gap between this uniform escape rate and the one-sided local escape rate of Dvoretsky and Erdos and the two-sided local escape rate of Jain and Taylor suggest the study of certain sets of times of slow one- or two-sided escape. The Hausdorff dimension of these exceptional sets is computed. The results are proved for a broad class of strictly stable processes.

Large deviation theorems at the Donsker-Varadhan level-three type are established for certain classes of stationary processes. Proceeding from more general to more specific assumptions allows us to describe the rate function more explicitly.

The main theorems in this paper prove uniform large deviation properties for the empirical measure and the multivariate empirical measure of a Markov chain that takes values in a complete separable metric space. One contribution of the paper is that, in contrast to previous large deviation results for the empirical measure, we do not assume that the transition probability of the Markov chain has a density with respect to a reference measure.

Consider the recursive, stochastic algorithm $X^\varepsilon_{n+1} = X^\varepsilon_n + \varepsilon b(X^\varepsilon_n, \xi_n)$, where $\{\xi_n\}$ is a random process and $X^\varepsilon_n$ lives in $\mathbb{R}^d$. Algorithms of this type arise frequently in applications in control and communications, as well as elsewhere. In the study of the important long term behavior of such recursive algorithms the "large deviations" behavior of the system, which describes the asymptotics of the order 1 deviations of the system from its "mean" trajectory as $\varepsilon$ tends to 0, plays a central role. Typical systems arising in communication theory and control often use complicated forcing terms involving correlated and state dependent noises and forcing terms with discontinuities. This paper presents a general approach for proving large deviations type theorems for such systems. The problem of proving such a theorem is considered first for the general case of a stochastic process with Lipschitz continuous sample paths. The assumptions are stated in terms of the conditional distribution of time increments of the process. After giving the proof in this general framework, we give several examples (in both continuous and discrete time) of driving terms that satisfy the hypotheses. The results are subsequently extended to a "projected" version of the discrete time model. The paper concludes with an application of the results to an automatic routing mechanism.

We consider noninteracting infinite particles, each of which follows a diffusion with generator $L \equiv (D^2 + D)/2$. The presence of many invariant distributions makes the situation radically different from the more familiar case where strong ergodicity assumptions are made. Explicit large deviation rates for the empirical density are obtained. The dependence of the rates on the initial distribution is strong and can be seen clearly. Some variational formulas for the scattering data associated with $L$ are also obtained.

Let $\eta_t$ be the (basic) voter model on $\mathbb{Z}^d$. We consider the occupation time functionals $\int^t_0 f(\eta_s)ds$ for certain functions $f$ and initial distributions. The first result is a pointwise ergodic theorem in the case $d = 2$, extending the work of Andjel and Kipnis. The second result is a central limit type theorem for $f(\eta) = \eta(0)$ and initial distributions: (i) $\delta_\eta$, for a class of states $\eta, d \geq 2$, and (ii) $\nu_\theta$, the extremal invariant measures, $d \geq 3$.

In this paper we complete the work started in part I. We show that if $\sigma_N$ is the time that the contact process on $\{1,\ldots, N\}$ first hits the empty set, then for $\lambda > \lambda_c$ (the critical value for the process on $Z$) there is a positive constant $\gamma(\lambda)$ so that $(\log \sigma_N)/N\rightarrow\gamma(\lambda)$ in probability as $N\rightarrow\infty$. We also give a new simple proof that $\sigma_N/E\sigma_N$ converges to a mean one exponential. The keys to the proof of the first result are a "planar graph duality" for the contact process and an observation of J. Chayes and L. Chayes that exponential decay rates for connections in strips approach the decay rates in the plane as the width of the strip goes to $\infty$.

We extend to general $p$-stable processes the lower bound that Marcus and Pisier established for strongly stationary $p$-stable processes. This bound also extends the results of the author on general Gaussian processes.

We examine the paths of the stable fields that are the analogs of index-$\beta$ Gaussian fields. We find Holder conditions on their paths and find the Hausdorff dimension of the image, graph and level sets when we have local nondeterminism, generalizing the Gaussian results.

A representation for the distribution of the trimmed sum of vector-valued random variables is obtained, generalising a one-dimensional formula. The trimming is with respect to observations falling outside a fixed family of sets, e.g., spheres. Asymptotic normality of the heavily trimmed sum, when normed and centered in different ways, is proved, and rates of convergence are given for some cases.

A thorough description of the almost sure behavior of weighted uniform quantile processes is given. This includes analogues of nearly all known results for weighted uniform empirical processes, such as the James functional law of the iterated logarithm and the Csaki results on the supremum of the standardized empirical process. Subject to the usual regularity conditions, our results extend to the nonuniform quantile process. Also, in the process of obtaining our results, we derive an extension of a theorem of Kiefer, which is likely to be of independent interest.

Let $X^0_1, \ldots, X^0_N$ be integer-valued random variables and let $a_1, \ldots, a_N$ be (fixed) nonzero vectors. We introduce the notion of coarseness of a discrete distribution and obtain upper bounds on the coarseness of the distribution of $S = \Sigma X^0_ia_i$ by comparison with the case $a_i \equiv a$. The bounds derived are seen to be tight and to apply for example when $S$ is formed (a) from independent summands or (b) by using any of a large family of sampling schemes. We show how such bounds can easily and efficiently substitute for use of Berry-Esseen theorems and other analytical methods in applications.

Let $X_1, \ldots, X_n$ be independent and identically distributed random variables with zero mean and unit variance. It is shown that the random bootstrap approximation to the distribution of $S \equiv n^{-1/2} \sum_jX_j$, converges to normality at precisely the same rate as $n^{-3/2}|\sum_jX^3_j| + n^{-2}\sum_jX^4_j$ converges to 0, up to terms of smaller order than $n^{-1/2}$. This result is used to explore properties of the bootstrap approximation under conditions weaker than existence of finite third moment. In most cases of that type it turns out that the bootstrap approximation to the distribution of $S$ is asymptotically equivalent to the normal approximation, so that the numerical expense of calculating the bootstrap approximation would not be justified. There also exist circumstances where the third moment is "almost" finite, yet the bootstrap approximation is asymptotically much worse than the simpler normal approximation. Necessary and sufficient conditions are given for a one-term Edgeworth expansion of the bootstrap approximation.

We give uniform rates of convergence in the central limit theorem for associated processes with finite third moment. No stationarity is required. Using a coefficient $u(n)$ which describes the covariance structure of the process, we obtain a convergence rate $O(n^{-1/2}\log^2n)$ if $u(n)$ exponentially decreases to 0. An example shows that such a rate can no longer be obtained if $u(n)$ decreases only as a power.

The main result of the present paper is a sharp nonuniform bound on the rate of convergence to normality in the central limit theorem for martingales having finite moments of order $2 + 2\delta$ for some $0 < \delta < \infty$. A nonuniform bound on the rate for convergence to mixtures of normal distributions is obtained as a consequence.

If $\mathbf{M}$ is a fixed $d \times d$ complex-valued matrix, then the eigenvalues of $\mathbf{M}'$, the conjugate transpose of $\mathbf{M}$, are the complex conjugates of the eigenvalues of $\mathbf{M}$, with the same multiplicities, and if $\mathbf{M}$ is upper block triangular, the eigenvalues of $\mathbf{M}$ are the eigenvalues of the diagonal blocks, and the multiplicities add. We shall show that if $\{\mathbf{M}(j)\}$ is a stationary, ergodic, time-reversible sequence taking values in the $d \times d$ complex matrices, then similar properties hold for the Lyapunov exponents of $\{\mathbf{M}(j)\}$.

Let $W = (X_{ij}; 1 \leq i, j < \infty)$ be an infinite matrix. Suppose $W$ is symmetric, entries on the diagonal are $\operatorname{iid}$, entries off the diagonal are $\operatorname{iid}$ and they are independent. Then it is proved that the necessary and sufficient conditions for $\lambda_{\max}((1/\sqrt{n})W_n) \rightarrow a \mathrm{a.s.}$ are (1) $E(X^+_{11})^2 < \infty$; (2) $EX^4_{12} < \infty$; (3) $EX_{12} \leq 0$; (4) $a = 2\sigma, \sigma^2 = EX^2_{12}$. Here $W_n = (X_{ij}; 1 \leq i, j \leq n)$.

The concept of random cell complexes, generalised sets and their mean sets in $R^d$ are introduced. Under stationarity conditions various relations between associated geometrical quantities are derived.

Let $X_i, 1 \leq i < \infty$, denote independent random variables with values in $\mathbb{R}^d, d \geq 2$, and let $M_n$ denote the cost of a minimal spanning tree of a complete graph with vertex set $\{X_1, X_2, \ldots, X_n\}$, where the cost of an edge $(X_i, X_j)$ is given by $\psi(|X_i - X_j|)$. Here $|X_i - X_j|$ denotes the Euclidean distance between $X_i$ and $X_j$ and $\psi$ is a monotone function. For bounded random variables and $0 < \alpha < d$, it is proved that as $n\rightarrow\infty$ one has $M_n \sim c(\alpha, d)n^{(d - \alpha)/d} \int_{R^d} f(x)^{(d-\alpha)/d} dx$ with probability 1, provided $\psi(x) \sim x^\alpha$ as $x\rightarrow 0$. Here $f(x)$ is the density of the absolutely continuous part of the distribution of the $\{X_i\}$.

Consider the lattice $Z^d, d \geq 1$, together with a stochastic black-white coloring of its points and on it a random walk that is independent of the coloring. A local scenery perceived at a given time is a pattern of colors seen by the walker in a finite box around his current position. Under weak assumptions on the probability distributions governing walk and coloring, we prove asymptotic independence of local sceneries perceived at times 0 and $n$, in the limit as $n\rightarrow\infty$, and at times 0 and $T_k$, in the limit as $k \rightarrow \infty$, where $T_k$ is the random $k$th hitting time of a black point. An immediate corollary of the latter result is the convergence in distribution of the interarrival times between successive black hits, i.e., of $T_{k+1} - T_k$ as $k\rightarrow\infty$. The limit distribution is expressed in terms of the distribution of the first hitting time $T_1$. The proof uses coupling arguments and ergodic theory.

Hoeffding's lemma gives an integral representation of the covariance of two random variables in terms of the difference between their joint and marginal probability functions, i.e., $\operatorname{cov}(X, Y) = \int^\infty_{-\infty} \int^\infty_{-\infty} \{P(X > x, Y > y) - P(X > x)P(Y > y)\} dx dy.$ This identity has been found to be a useful tool in studying the dependence structure of various random vectors. A generalization of this result for more than two random variables is given. This involves an integral representation of the multivariate joint cumulant. Applications of this include characterizations of independence. Relationships with various types of dependence are also given.

For each $\delta > 0$ there is a parabolic operator in the half-plane $R_x \times R^+_t$ whose parabolic measure is supported by a boundary set of dimension $< \delta$.

We consider nonnegative infinitely divisible random variables whose Levy measures are either absolutely continuous or supported by the integers. Necessary conditions are found ensuring that such distributions are $\log$-concave or $\log$-convex.

We study the solution $\xi_t$ to an elliptic stochastic equation and the solution $(\xi_t, \eta_t)$ to a lift of this equation, under a condition on the behaviour of $\xi_t$. We obtain a "conditional" equation which gives the conditional behaviour, and we use it to deduce a conditioned version of the Stroock-Varadhan support theorem.

It is shown that the algorithms for determining the generating function and prediction error matrix of multivariate stationary stochastic processes developed by Weiner and Masani (1957, 1958) and later by Masani (1960) will work in some more general setting.