Let $\{X_k\}$ be an i.i.d. sequence and define $S_n = X_1 + \cdots + X_n$. The problem is to determine for a given sequence $\{\beta_n\}$ whether $P\{|S_n| \leq \beta_n \mathrm{i.o.}\}$ is 0 or 1. A history of the problem is given along with two new results for the case when $P\{X_1 \geq 0\} = 1$: (a) An integral test that solves the problem in case the summands satisfy Feller's condition for stochastic compactness of the appropriately normalized sums and (b) necessary and sufficient conditions for a sequence $\{\beta_n\}$ to exist such that $\lim \inf S_n/\beta_n = 1$ a.s.

By producing a finite-box criterion for survival in a slab, we show that the critical contact process dies out and that, in two and more dimensions, the critical parameter value coincides with the limit of slab thresholds. Using the techniques developed in this paper, one may obtain the complete convergence and shape theorems for contact processes in all dimensions and for all infection rates. Our results apply also to the discrete-time analogue of the contact process, viz., oriented percolation.

A strong stationary time for a Markov chain $(X_n)$ is a stopping time $T$ for which $X_T$ is stationary and independent of $T$. Such times yield sharp bounds on certain measures of nonstationarity for $X$ at fixed finite times $n$. We construct an absorbing dual Markov chain with absorption time a strong stationary time for $X$. We relate our dual to a notion of duality used in the study of interacting particle systems. For birth and death chains, our dual is again birth and death and permits a stochastic interpretation of the eigenvalues of the transition matrix for $X$. The duality approach unifies and extends the analysis of previous constructions and provides several new examples.

We study a class of particle systems which includes finite-range spin systems and combinations of those systems with stirring processes. We give sufficient conditions for ergodicity of the processes. The method is based on a graphical representation of the system and construction of a "generalized dual process."

We consider one dimensional, attractive stochastic Ising models with finite range interactions in which the particles move according to the stirring process. We prove that the process is exponentially ergodic provided the migration rate is sufficiently small.

The probability $P\lbrack 0 \leftrightarrow x \rbrack$ of connection of 0 to $x$ by a path of occupied bonds for Bernoulli percolation at density $p$ below the critical point is known to decay exponentially for each direction $x \in \mathbb{Z}^d$, in that $P\lbrack 0 \leftrightarrow nx \rbrack \approx e^{-n\sigma g(x)}$ as $n \rightarrow \infty$ for some $\sigma > 0$ and $g(x)$ of order $\|x\|$. This approximation is also an upper bound: $P\lbrack 0 \leftrightarrow x \rbrack \leq e^{-\sigma g(x)}$ for all $x$. Here a complementary power-law lower bound is established for $d = 2$ and 3: $P\lbrack 0 \leftrightarrow x \rbrack \geq c\|x\|^{-r} e^{-\sigma g(x)}$ for some $r = r(d)$ and $c = c(p,d)$.

A Russo-Seymour-Welsh (RSW) theorem is established for continuum percolation on $\mathbb{R}^2$. The equality of various definitions of critical densities for the continuum percolation on $\mathbb{R}^2$ is deduced as an application of the RSW theorem. It is also shown that various notions of the size of a cluster yield the same notion of critical density.

Let $W(t, \varepsilon)$ be the $\varepsilon$-Wiener sausage, i.e., the $\varepsilon$-neighborhood of the trace of the Brownian motion up to time $t$. It is shown that the results of Donsker and Varadhan on the behavior of $E(\exp(-\nu|W(t, \varepsilon)|)), \nu > 0$, remain true if $\varepsilon$ depends on $t$ and converges to 0 with a certain rate.

We prove infinite-time extensions of invariance principles for certain random walks with essentially compact state spaces. The extensions are uniform-like in time since they use the $\bar{d}$-metric of the Bernoulli theory and imply the classical results. These are then generalized to couplings involving an isomorphism between the processes. In general a Doeblin-type condition is needed to hold for the walks but relaxation of this is indicated.

Let $L_\varepsilon = \frac{1}{2}\Delta + \varepsilon b \cdot \nabla$ in $R^d, d \geq 3$, generate a recurrent diffusion for each $\varepsilon > 0$, where $b \in C^\alpha(R^d)$, and let $D \subset R^d$ be an exterior domain. Then by the recurrence assumption, for each $\psi \in C(\partial D)$, there exists a unique solution in the class of bounded solutions to the Dirichlet problem $L_\varepsilon u_\varepsilon = 0$ in $D$ and $u_\varepsilon = \psi$ on $\partial D$. On the other hand, by the transience of $d$-dimensional Brownian motion, there is no uniqueness in the class of bounded solutions for the Dirichlet problem $\frac{1}{2} \Delta u = 0$ in $D$ and $u = \psi$ on $\partial D$. Since the Martin boundary at $\infty$ for Brownian motion consists of a single point, uniqueness is obtained by adding the condition $\lim_{|x|\rightarrow\infty} u(x) = c$. We show that $u_0(x) \equiv \lim_{\varepsilon\rightarrow 0} u_\varepsilon(x)$ exists and satisfies $\frac{1}{2}\Delta u_0 = 0$ in $D, u_0 = \psi$ on $\partial D$ and $\lim_{|x|\rightarrow\infty} u_0(x) = c$, where $c$ is given as follows. Let $P^h_x$ denote the measure associated with Doob's conditioned Brownian motion conditioned to exit $D$ at $\partial D$ rather than at $\infty$. Let $\tau = \inf\{t \geq 0: X(t) \in \partial D\}$ and define the harmonic measure $u^h_x(dy) = P^h_x(X(\tau) \in dy)$. Then $\mu^h_\infty \equiv \lim_{|x|\rightarrow\infty} \mu^h_x$ exists and $c = \int_{\partial D}\psi(y)\mu^h_\infty(dy)$. We also show that the energy integral $\int_D|\nabla u|^2 dx$, when varied over all bounded functions $u \in W^{1,2}_{\operatorname{loc}}(D)$ which satisfy $u = \psi$ on $\partial D$, takes on its minimum uniquely at $u_0$.

Let $X$ be a strong Markov process with potential kernel $U$. We show that if $(\nu_n)$ and $\mu$ are measures on the state space of $X$ such that $\nu_1 U \leq \nu_2 U \leq \cdots \leq \mu U$, then there is a decreasing sequence $(T_n)$ of randomized stopping times such that $\nu_n$ is the law of $X_{T_n}$ when the initial distribution of $X$ is $\mu$.

Let $X$ be a transient right (Markov) process on a compact metric space including a death point. Let $\mu$ and $\nu_n$ be finite measures whose potentials satisfy $\mu U \geq \cdots \geq \nu_nU \geq \cdots \geq \nu_1 U$. We prove that there exists a right-continuous stochastic process $Y = (\tilde\Omega, \mathscr{M}, \tilde\mathscr{M}_t, Y_t Q)$ that is a version of $X$ with initial measure $\nu_\infty(\cdot) = Q(Y_0 \in \cdot)$ and in which there are $(\tilde\mathscr{M}_t)$-stopping times $\tilde{\tau}_n \downarrow 0$ with $Q(Y(\tilde{\tau}_n) \in \cdot, \tilde{\tau}_n < \infty) = \nu_n(\cdot)$. Furthermore, a canonical representation of $Y$ and $(\tilde{\tau}_n)$ is given in which one has a better understanding of the tail behavior of the sequence $\tilde{\tau}_n$. Based on this representation an open question is posed whose answer in the positive would permit defining in $X$, assuming it admits a continuous real random variable independent of the path, decreasing stopping times $T_n$ such that $P^\mu(X(T_n) \in \cdot, T_n < \infty) = \nu_n(\cdot)$. These $T_n$ would satisfy the Markov property $T_n = T_{n+1} + S_n \circ \theta(T_{n+1}), S_n$ a stopping time linking $\nu_n$ and $\nu_{n+1}$. Fitzsimmons has now proved the existence of a desired decreasing sequence $T_n$ in $X$ for any given $\mu$ and $\nu_n$ as above, using a very different approach. His $T_n$, however, do not satisfy the Markov property.

Stochastic Volterra equations are studied where the coefficients $F(t, s, x)$ are random and adapted to $\mathscr{F}_{s\vee t}$ rather than the customary $\mathscr{F}_{s\wedge t}$. Such a hypothesis, which is natural in several applications, leads to stochastic integrals with anticipating integrands. We interpret these as Skorohod integrals, which generalize Ito's integrals to the case where the integrand anticipates the future of the Wiener integrator. We shall nevertheless construct an adapted solution, which is even a semimartingale if the coefficients are smooth enough.

The following generalization of Rosenthal's inequality was proved by Burkholder: $A^{-1}_p\{\|s(f)\|_p + \|d^\ast\|_p\} \leq \|f^\ast\|_p \leq B_p\{\|s(f)\|_p + \|d^\ast\|_p\},$ for all martingales $(f_n)$. It is known that $A_p$ grows like $\sqrt{p}$ as $p \rightarrow \infty$. In this paper we prove that the growth rate of $B_p$ as $p \rightarrow \infty$ is $p/\ln p$.

Let $X(t)$ be a strictly stable Levy process of index $1 < \alpha \leq 2$ and skewness index $|h| \leq 1$. Let $L^x_t$ be its local time and $L^\ast_t = \sup_x L^x_t$ the maximum local time. We show that $\lim_{\lambda \rightarrow + \infty} \lambda^{-\alpha} \log P(L^\ast_1 > \lambda) = -C_{\alpha h}$, where $C_{\alpha h}$ is a known constant. In the case that $X(t)$ is a standard Brownian motion, $C_{\alpha h} = 1/2$ and the result is due to Perkins.

We present a version of the Wentzell-Freidlin theory for Markov chains which includes random perturbations not only of deterministic motions but also of Markov chains. Some results for the continuous-time case are obtained as corollaries. In particular, by this method one can treat random perturbations of degenerate diffusions even when the large deviations principle fails.

Let $\{\alpha_n(t), 0 \leq t \leq 1\}$ and $\{\beta_n(s), 0 \leq s \leq 1\}$ denote the uniform empirical and quantile processes. We show that, for suitable sequences $A(n, \kappa_n)$ and $B(n, l_n)$, the tail empirical process $\{A(n, \kappa_n)\alpha_n(n^{-1}\kappa_nt), 0 \leq t \leq 1\}$ and the tail quantile process $\{B(n, l_n)\beta_n(n^{-1}l_n s), 0 \leq s \leq 1\}$ are almost surely relatively compact in appropriate topological spaces, where $0 \leq \kappa_n \leq n$ and $0 \leq l_n \leq n$ are sequences such that $\kappa_n$ and $l_n$ are $O(\log \log n)$ as $n \rightarrow \infty$. The limit sets of functions are defined through integral conditions and differ from the usual Strassen set obtained when $\kappa_n$ and $l_n$ are $\infty(\log \log n)$ as $n \rightarrow \infty$. Our results enable us to describe the strong limiting behavior of classical statistics based on the top extreme order statistics of a sample or on the empirical distribution function considered in the tails.

Let $\alpha_n$ be the empirical process of independent uniformly distributed random vectors on the unit square $I^2$. We study the asymptotic distribution of the random variable $\sup|\alpha_n(s,t)|/(s^\nu t^\mu L(s)G(s))$ when $\sup$ is taken over various subintervals of $I^2$. We show that in the case of $-\infty < \mu, \nu < 1/2$ the limit is given in terms of a two-time parameter Wiener process, and for $1/2 < \mu, \nu < \infty$ it is determined by a Poisson process.

In this paper, we give a simple condition ensuring that a Gaussian random function $X$ on a metric space $T$ with values in a Lusin topological vector space has a modification with continuous paths. This result extends previous results where $X$ was supposed to be stationary or have stationary increments. As in the stationary case, proof is based on Talagrand's theorem about the majorizing measures which permit us, if $E$ is a separable Banach space, to bound the law of the maximum on $T$ of the norm of $X$ in $E$.

It is shown that every symmetric $\alpha$-stable random variable $X, 0 < \alpha \leq 2$, has the property: For any $p$ and $q, 0 \leq h(\alpha) < q < p < \alpha$, there is a constant $s > 0$ such that $(E\|x + sXy\|^p)^{1/p} \leq (E\|x + Xy\|^q)^{1/q},$ for all $x$ and $y$ from a normed space. The quantity $h(\alpha)$ is identically 0 if $\alpha \leq 1$. It is strictly less than 1 for every $\alpha < 2$ which reveals the contrast to the Gaussian case in which $q > h(2) = 1$.

We present necessary and sufficient conditions for association and negative association of jointly $\alpha$-stable random variables, $0 < \alpha < 2$.

Let $\{X_t\}^\infty_{t=1}$ be a stationary Gaussian time series with zero mean, unit variance, absolutely summable autocorrelation function and at least once differentiable spectral density function which is strictly positive in $\lbrack 0, \pi \rbrack$. In this paper it is shown that, if $M_n$ denotes the maximum of the normalized periodogram of $\{X_1,\ldots, X_n\}$ over the interval $\lbrack 0, \pi \rbrack$, then, almost surely, \begin{equation*}\tag{1} \lim \inf_{n\rightarrow\infty} \lbrack M_n - 2 \log n + \log \log n \rbrack \geq 0\end{equation*} and \begin{equation*}\tag{2} \lim \sup_{n\rightarrow\infty} \lbrack M_n - 2 \log n - 2(\log n)^\delta \rbrack = -\infty\end{equation*} for any $\delta > 0$.

Let $R(n)$ be the largest integer for which the disc of radius $R(n)$ around the origin is covered by the first $n$ steps of a random walk. The main objective of the present paper is to obtain better estimates for the upper tail of the distribution of $R(n)$. For example, we show that there are constants $0 < \lambda_2 < \lambda_1 < \infty$ such that \begin{align*}\exp(-\lambda_1 z) &\leq \lim \inf_{n\rightarrow\infty} \mathbf{P}\big\{\frac{(\log R(n))^2}{\log n} > z\big\} \\ &\leq \lim \inf_{n\rightarrow\infty}\mathbf{P}\big\{\frac{(\log R(n))^2}{\log n} > z\big\} \leq \exp(-\lambda_2z).\end{align*}

Let $Z_1, Z_2,\ldots$ be jointly distributed random variables for which $\sup_k Z_k = \infty \mathrm{w.p.}1$ and let $t = t_a = \inf(n \geq 1: Z_n > a)$ and $R_a = Z_t - a$ for $a \geq 0$. Conditions under which $R_a$ has a limiting distribution as $a \rightarrow \infty$ are developed. These require that the finite dimensional, conditional distributions of the increments $Z_{t+k} - Z_t, k \geq 1$, converge to the finite dimensional distributions of a process for which the result is known, thus weakening the slow change condition in earlier work. The main result is applied to some sequences for which the limiting distributions are those of the partial sums of an exchangeable process. These include the Euclidean norms of a driftless random walk in several dimensions and sequences for which the conditional distribution of $Z_{n+1} - Z_n$ given the past has a limit $\mathrm{w.p.}1$ as $n \rightarrow \infty$.

A representation for products of finite nonnegative matrices is given in terms of products of stochastic matrices and as a result Markov chain arguments are used to derive ratio limit properties. In particular, we obtain necessary and sufficient conditions for weak ergodicity and give a probabilistic proof of the Coale-Lopez theorem. In the general case, there are several sequences of sets of partitions of the state space corresponding to an associated nonhomogeneous Markov chain which lead to a number of ratio product limits. Asymptotic column proportionality, characteristic of weak ergodicity, may occur only inside each sequence of sets with one possible exception.

Let $f: M \rightarrow N$ be a harmonic map between complete Riemannian manifolds $M$ and $N$, and suppose the Ricci curvature of $M$ is nonnegative definite, the sectional curvature of $N$ is nonpositive, and $N$ is simply connected. Then if $f$ has sublinear asymptotic growth, $f$ must be a constant map. This result was first proved analytically by S.-Y. Cheng. This paper describes a probabilistic proof under the same hypotheses.