We show that certain $d$-dimensional Markov processes $X(t), t\geq 0$, have the property that if $E$ is a closed subset of $R_+$ with sufficiently large Hausdorff dimension, then $X(E)$ has $k$-multiple points. This is applied directly to diffusions driven by stochastic differential equations and Levy processes with positive lower indices, solving problems posed by J. P. Kahane and S. J. Taylor.

We prove that if $G$ is a subgroup of the (time-change) symmetry group of a Markov process $X_t$ which is transitive and has a compact isotropy subgroup, then after a time change, $X_t$ becomes $G$-invariant. The symmetry groups of diffusion processes are discussed in more detail. We show that if the generator of $X_t$ is the Laplacian with respect to the intrinsic metric, then $X_t$ has the best invariance property.

We consider a class of stochastic processes of a type that was first introduced by Dubins and Freedman. These processes are indexed by the lines of descent through an infinite tree and take values in a space of sequences. Our main results concern necessary and sufficient conditions of a potential theoretic type for a subset of the state-space to be hit with positive probability by the sample paths of the process. We examine these conditions in some specific examples and also relate them to conditions expressed in terms of Hausdorff dimension. As well, we use similar techniques to investigate multiple points in the sample paths of the process.

This paper examines the question of when a two-parameter process $X$ of independent increments will have Levy's sharp Markov property relative to a given domain $D$. This property states intuitively that the values of the process inside $D$ and outside $D$ are conditionally independent given the values of the process on the boundary of $D$. Under mild assumptions, $X$ is the sum of a continuous Gaussian process and an independent jump process. We show that if $X$ satisfies Levy's sharp Markov property, so do both the Gaussian and the jump process. The Gaussian case has been studied in a previous paper by the same authors. Here, we examine the case where $X$ is a jump process. The presence of discontinuities requires a new formulation of the sharp Markov property. The main result is that a jump process satisfies the sharp Markov property for all bounded open sets. This proves a generalization of a conjecture of Carnal and Walsh concerning the Poisson sheet.

We briefly show an extension of inequalities of Burkholder and Gundy for linear Brownian motion to certain monotone functionals of the $d$-dimensional Brownian convex hull. Our results belong to a class of results that imply that Brownian hulls are much like the one-dimensional maximal process.

A planar domain whose boundary $E$ lies in the real line is called a Denjoy domain. In this article we consider some geometric properties of Brownian motion in such a domain. The first result is that if $E$ has zero length $(|E| = 0)$, then there is a set $F \subset E$ of full harmonic measure such that every Brownian path which exits at $x \in F$ hits both $(-\infty, x)$ and $(x,\infty)$ with probability 1, verifying a conjecture of Burdzy. Next we show that if $\dim(E) < 1$, then almost every Brownian path forms infinitely many loops separating its exit point from $\infty$ and we give an example to show $\dim(E) < 1$ cannot be replaced by $|E| = 0$.

It is proven that, for any deterministic $L^2\lbrack 0,1\rbrack$ function $\phi(t)$, $E\bigg(\exp\int^1_0\phi(t)dw_t\bigg\arrowvert \|w\| < \varepsilon\bigg) \rightarrow 1\,\text{as}\,\varepsilon \rightarrow 0,$ where $w_t$ is a standard Brownian motion and $\|\cdot\|$ is any "reasonable" norm on $C_0\lbrack 0,1\rbrack$. Applications to the computation of Onsager-Machlup functionals are pointed out.

Let $X_t$ be Brownian motion on a Riemannian manifold $M$ started at $m$ and let $T$ be the first time $X_t$ exits a normal ball about $m$. The first exit time $T$ for $M = S^3 \times H^3$ has the same distribution as the first exit time for $M = \mathbf{R}^6$. For $M = S^3 \times H^3, T$ and $X_T$ are independent random variables.

Let $X_{\bar n}, \bar{n} \in \mathbb{N}^d$, be a field of independent real random variables, where $\mathbb{N}^d$ is the $d$-dimensional lattice. In this paper, the law of the iterated logarithm is established for such a field of random variables. Theorem 1 brings into focus a connection between a certain strong law of large numbers and the law of the iterated logarithm. A general technique is developed by which one can derive the strong law of large numbers and the law of the iterated logarithm, exploiting the convergence rates in the weak law of large numbers in Theorem 2. In Theorem 3, we use Gaussian randomization techniques to obtain the law of the iterated logarithm which generalizes Wittmann's result.

The Kolmogorov law of the iterated logarithm fails when the boundedness condition on the increments is relaxed. In this paper, we consider this in the martingale setting and establish a lower bound, extending a result known in the independent case.

The tail empirical process is defined to be for each $n \in \mathbb{N}, w_n(t) = (n/k_n)^{1/2}\alpha_n(tk_n/n), 0 \leq t \leq 1$, where $\alpha_n$ is the empirical process based on the first $n$ of a sequence of independent uniform (0,1) random variables and $\{k_n\}^\infty_{n=1}$ is a sequence of positive numbers with $k_n/n \rightarrow 0$ and $k_n \rightarrow \infty$. In this paper a complete description of the almost sure behavior of the weighted empirical process $a_n\alpha_n/q$, where $q$ is a weight function and $\{a_n\}^\infty_{n=1}$ is a sequence of positive numbers, is established as well as a characterization of the law of the iterated logarithm behavior of the weighted tail empirical process $w_n/q$, provided $k_n/\log\log n \rightarrow \infty$. These results unify and generalize several results in the literature. Moreover, a characterization of the central limit theorem behavior of $w_n/q$ is presented. That result is applied to the construction of asymptotic confidence bands for intermediate quantiles from an arbitrary continuous distribution.

Consider the inhomogeneous contact process on $Z^1$ with recovery rate $\delta(k)$ at site $k$ and infection rates $\lambda(k)$ and $\rho(k)$ at site $k$ due to the presence of infected neighbors at $k - 1$ and $k + 1$ respectively. A special case of the main result in this paper is the following: Suppose that the environment is chosen in such a way that the $\delta(k)$'s, $\lambda(k)$'s and $\rho(k)$'s are all mutually independent, with the $\delta(k)$'s having a common distribution, and the $\lambda(k)$'s and $\rho(k)$'s having a common distribution. Then the process survives if $E\frac{\delta(\lambda + \rho + \delta)}{\lambda\rho} < 1,$ while the right edge $r_t$ of the process with initial configuration $\cdots$ 111000 $\cdots$ satisfies $\lim\sup_{t\rightarrow\infty}r_t = +\infty$ if $E\log\frac{\delta(\lambda + \rho + \delta)}{\lambda\rho} < 0.$ If the environment is deterministic and periodic with period $p$, we prove survival if $\Pi^p_{k=1}\frac{\delta(k)\lbrack\lambda(k) + \rho(k - 1) + \delta(k)\rbrack}{\lambda(k)\rho(k - 1)} < 1$ and $\Pi^p_{k=1}\frac{\delta(k - 1)\lbrack\lambda(k) + \rho(k - 1) + \delta(k - 1)\rbrack}{\lambda(k)\rho(k - 1)} < 1.$

The sexual reproduction process is a reaction-diffusion type interacting particle system which exhibits phase transition. In this paper an upper bound is given for the critical value of the reaction rate and the equilibrium density is determined for all noncritical values of the reaction rate. The method of renormalized bond construction is the basis of our proof. Results for the hydrodynamic limit of the process are employed and some results for the associated nonlinear partial differential equation are developed to build the elements of this construction.

Let $\{Z_n\}$ be a $p$ type positively regular nonsingular critical branching process with mean matrix $M$. If $\nu$ is a right eigenvector of $M$ for the eigenvalue 1 and $Y_n = Z_n \cdot \nu$, and if $M_n = \max_{0\leq j\leq n}Y_j$, then it is shown that under second moments $(\log n)^{-1}E_\mathbf{i}M_n \rightarrow \mathbf{i \cdot v}$, where $E_\mathbf{i}$ denotes starting with $Z_0 = \mathbf{i}$ and $\cdot$ denotes inner product. This is an extension of the result for the single type case obtained by Athreya in 1988.

An analogue of the Lai-Siegmund nonlinear renewal theorem is proved for processes of the form $S_n + \xi_n$, where $\{S_n\}$ is a Markov random walk. Specifically, $Y_0,Y_1,\cdots$ is a Markov chain with complete separable metric state space; $X_1,X_2,\cdots$ is a sequence of random variables such that the distribution of $X_i$ given $\{Y_j, j \geq 0\}$ and $\{X_j, j \neq i\}$ depends only on $Y_{i - 1}$ and $Y_i; S_n = X_1 + \cdots + X_n$; and $\{\xi_n\}$ is slowly changing, in a sense to be made precise below. Applications to sequential analysis are given with both countable and uncountable state space.

The distributions of random walk quantities like ascending ladder heights and the maximum are shown to be phase-type provided that the generic random walk increment $X$ has difference structure $X = U - T$ with $U$ phase-type, or the one-sided assumption of $X_+$ being phase-type is imposed. As a corollary, it follows that the stationary waiting time in a GI/PH/1 queue with phase-type service times is again phase-type. The phase-type representations are characterized in terms of the intensity matrix $\mathbf{Q}$ of a certain Markov jump process associated with the random walk. From an algorithmic point of view, the fundamental step is the iterative solution of a fix-point problem $\mathbf{Q} = \psi(\mathbf{Q})$, and using a coupling argument it is shown that the iteration typically converges geometrically fast. Also, a variant of the classical approach based upon Rouche's theorem and root-finding in the complex plane is derived, and the relation between the approaches is shown to be that $\mathbf{Q}$ has the Rouche roots as its set of eigenvalues.

Consider a single-server queue with units served in order of arrival for which we can define a stationary distribution (equilibrium distribution) of the vector of the waiting time and the queue size. Denote this vector by $(w(\rho), l(\rho))$, where $\rho < 1$ is the traffic intensity in the system when it is in equilibrium and $\lambda_\rho$ is the intensity of the arrival stream to this system. Szczotka has shown under some conditions that $(1 - \rho)(l(\rho) - \lambda_\rho w(\rho)) \rightarrow_p 0$ as $\rho\uparrow 1$ (in heavy traffic). Here we will show under some conditions that $\sqrt{1 - \rho}(l(\rho) - \lambda_\rho w(\rho)) \rightarrow_D bN\sqrt{M}$ as $\rho \uparrow 1$, where $N$ and $M$ are mutually independent random variables such that $N$ has the standard normal distribution and $M$ has an exponential distribution while $b$ is a known constant.

Let $\{S_n\}$ be a random walk on the integers with negative drift, and let $A_n = \{S_k \geq 0, 1 \leq k \leq n\}$ and $A = A_\infty$. Conditioning on $A$ is troublesome because $P(A) = 0$ and there is no natural sigma-field of events "like" $A. A$ natural definition of $P(B\mid A)$ is $\lim_{n\rightarrow\infty}P(B\mid A_n)$. The main result here shows that this definition makes sense, at least for a large class of events $B$: The finite-dimensional conditional distributions for the process $\{S_k\}_{k\geq 0}$ given $A_n$ converge strongly to the finite-dimensional distributions for a measure $\mathbf{Q}$. This distribution $\mathbf{Q}$ is identified as the distribution for a stationary Markov chain on $\{0,1,\ldots\}$.

Let $T_r$ be the first time a sum $S_n$ of nondegenerate i.i.d. random vectors leaves the sphere of radius $r$. The spheres are determined by some given norm on $\mathbb{R}^d$ which need not be the Euclidean norm. As a particular case of our results, we obtain, for mean-zero random vectors and each $0 < p < \infty$ and $0 \leq q < \infty$, necessary and sufficient conditions on the distribution of the summands to have $E(\|S_{T_r}\| - r)^p = O(r^q)$ as $r \rightarrow \infty$. We also characterize tightness of the family $\{\|S_{T_r}\| - r\}$ and obtain other related results on the rate of growth of $\|S_{T_r}\|$. In particular, we obtain a simple necessary and sufficient condition for $\|S_{T_r}\|/r \rightarrow_p 1$.

Let $X_1,X_2,\ldots$ be independent random variables such that $X_j$ has distribution $F_{\sigma(j)}$, where $\sigma(j) = 1$ or 2, and the distributions $F_i$ have mean 0. Assume that $F_i$ has a finite $q_i$th moment for some $1 < q_i < 2$. Let $S_n = \sum^n_{j=1}X_j$. We show that if $q_1 + q_2 > 3$, then $\lim\sup P\{S_n > 0\} > 0$ and $\lim\sup P\{S_n < 0\} > 0$ for each sequence $\{\sigma(j)\}$ of ones and twos.

We prove that in continuous time, the extremal elements of the set of adapted random measures on $\mathbb{R}^2_+$ are Dirac measures, assuming the underlying filtration satisfies the conditional qualitative independence property. This result is deduced from a theorem in discrete time which provides a correspondence between adapted random measures on $\mathbb{N}^2$ and two-parameter randomized stopping points in the sense of Baxter and Chacon. As an application we show the existence of optimal stopping points for upper semicontinuous two-parameter processes in continuous time.

We discuss universal schemes for portfolio selection. When such a scheme is used for investment in a stationary ergodic market with unknown distribution, the compounded capital will grow with the same limiting rate as could be achieved if the infinite past and hence of the distribution of the market were known to begin with. By specializing the market to a Kelly horse race, we obtain a universal scheme for gambling on a stationary ergodic process with values in a finite set. We point out the connection between universal gambling schemes and universal modeling schemes that are used in noiseless data compression. We also discuss a universal prediction scheme to learn, from past experience, the conditional distribution given the infinite past of the next outcome of a stationary ergodic process with values in a Polish space. This generalizes Ornstein's scheme for finite-valued processes. Although universal prediction schemes can be used to obtain universal gambling and portfolio schemes, they are not necessary.

We establish connections between superdiffusion processes and one class of nonlinear parabolic differential equations. Analytic results due to Brezis and Friedman, Baras and Pierre and others are used to investigate the graphs of superdiffusions. A survey of the literature and general comments are presented in Section 4.

We consider semilinear parabolic PDE's with fast oscillating boundary conditions. The central limit theorem and limit theorems for probabilities of large deviations for the solutions of such problems are proved.

Strong moderate deviation theorems are concerned with relative errors in the tails caused by replacing the exact distribution function by its limiting distribution function. A new approach for deriving such theorems is presented using strong approximation inequalities. In this way a strong moderate deviation theorem is obtained for statistics of the form $T(\alpha_n)$, where $T$ is a sublinear functional and $\alpha_n$ is the empirical process. The basic theorem is also applied on linear combinations of order statistics, leading to a substantial improvement of previous results.

The large deviation principle for the empirical field of a stationary $\mathbb{Z}^d$-indexed random field is proved under strong mixing dependence assumptions. The strong mixing coefficients considered allow us to separate the ratio-mixing condition used in the literature into a part directly responsible for the (nonuniform) large deviation principle and another one, which is used when the state space is noncompact. Results are applied to obtain variants of recent large deviation theorems for Markov chains and for Gibbs fields. The proofs are based on a new criterion for the large deviation principle which is stated in Appendix C.

Let $\{\Gamma(t), t \in \mathbb{R}\}$ be a Banach space $\mathscr{B}$-valued stochastic process. Let $P$ be the probability measure generated by $\Gamma(\cdot)$. Assume that $\Gamma(\cdot)$ is $P$-almost surely continuous with respect to the norm $\| \|$ of $\mathscr{B}$ and that there exists a positive nondecreasing function $\sigma(a), a > 0$, such that $P\{\|\Gamma(t + a) - \Gamma(t)\| \geq x\sigma(a)\} \leq K \exp(-\gamma x^\beta)$ with some $K, \gamma, \beta > 0$. Then, assuming also that $\sigma(\cdot)$ is a regularly varying function at zero, or at infinity, with a positive exponent, we prove large deviation results for increments like $\sup_{0\leq t\leq T-a}\sup_{0\leq s\leq a}\|\Gamma(t + s) - \Gamma(t)\|$, which we then use to establish moduli of continuity and large increment estimates for $\Gamma(\cdot)$. One of the many applications is to prove moduli of continuity estimates for $l^2$-valued Ornstein-Uhlenbeck processes.

A notion of $U$-exponents of a probability measure on a linear space is introduced. These are bounded linear operators and it is shown that the set of all $U$-exponents forms a Lie wedge for full measures on finite-dimensional spaces. This allows the construction of $U$-exponents commuting with the symmetry group of a measure in question. Then the set of all commuting exponents is described and elliptically symmetric measures are characterized in terms of their Fourier transforms. Also, self-decomposable measures are identified among those which are operator-self-decomposable. Finally, $S$-exponents of infinitely divisible measures are discussed.

We study the positive parabolic functions of the Ornstein-Uhlenbeck operator on an abstract Wiener space $E$ using the approach developed by Dynkin. This involves first proving a characterization of the entrance space of the corresponding Ornstein-Uhlenbeck semigroup and deriving an integral representation for an arbitrary entrance law in terms of extreme ones. It is shown that the Cameron-Martin densities are extreme parabolic functions, but that if $\dim E = \infty$, not every positive parabolic function has an integral representation in terms of those (which is in contrast to the finite-dimensional case). Furthermore, conditions for a parabolic function to be representable in terms of Cameron-Martin densities are proved.

Let $X^{(i)}_j, i = 1,\ldots, k; j \in \mathbf{N}$, be independent $d$-dimensional random vectors which are identically distributed for each fixed $i = 1,\ldots, k$. We give a sufficient condition for almost sure convergence of a sequence $T_{n_1,\ldots, n_k}$ of statistics based on $X^{(i)}_j i = 1,\ldots, k; j = 1, \ldots, n_i$, which are symmetric functions of $X^{(i)}_1,\ldots, X^{(i)}_{n_i}$ for each $i$ and do not change too much when variables are added or deleted. A key auxiliary tool for proofs is the Efron-Stein inequality. Applications include strong limits for certain nearest neighbor graph statistics, runs and empty blocks.