The asymptotic distributions for large times of a variety of additive functionals of planar Brownian motion $Z$ are derived. Associated with each point in the plane, and with the point infinity, there is a complex Brownian motion governing the asymptotic behavior of windings of $Z$ close to that point. An independent Gaussian field over the plane governs fluctuations in local occupation times of $Z$, while a further independent family of complex Brownian sheets governs finer features of the windings of $Z$. These results unify and extend earlier results of Kallianpur and Robbins, Spitzer, Kasahara and Kotani, Messulam and the authors.

Let $X$ be a standard two-dimensional Brownian motion. There exists a.s. $t \in (0, 1)$ such that $X(\lbrack 0, t)) \cap X((t, 1 \rbrack) = \varnothing$. It follows that $X(\lbrack 0, 1 \rbrack)$ is not homeomorphic to the Sierpinski carpet a.s.

The Onsager-Machlup function, namely the fictitious density of diffusion paths in function space, is considered, where the density is evaluated around non $C^2$ curves, thus extending earlier results. The extension holds also for the case of diffusions evolving on a manifold.

A process $(X_t)$ is equivalent to an Ornstein-Uhlenbeck process if and only if $e^{-\lambda t}f(X_t)$ is a martingale for every $f \geq 0$ on $\mathbb{R}^d$ such that $\Delta f(x) - \langle x, \nabla f(x)\rangle = \lambda f(x)$.

Let $W(t)$ for $0 \leq t < \infty$ be a standard Wiener process, suppose $0 < a_T \leq T$ for $T > 0$, and let $d(T, t) = \{2t\lbrack\log(T/t) + \log \log t \rbrack\}^{1/2}$. Quantities such as $\lim, \inf_{T \rightarrow \infty} \sup_{a_T \leq t \leq T} \frac{W(T) - W(T - t)}{d(T,t)},$ $\lim, \inf_{T \rightarrow \infty} \sup_{\substack{0 \leq t \leq T - a_T\\0 \leq s \leq a_T}} \frac{|W(t + s) - W(t)|}{d(t + a_T, a_T)}$ and $\lim, \inf_{T \rightarrow \infty} \sup_{\substack{0 \leq u < v \leq T\\a_T \leq v - u}} \frac{|W(v) - W(u)|}{d(v, v - u)}$ are investigated.

A particle system in Euclidean space is considered where the particles are subject to spatial motion according to a symmetric stable law and to a critical branching law in the domain of attraction of a stable law. The "branching intensity" may be position-dependent (varying medium) or be given by a realization of a random field (random medium). It is shown that under natural assumptions the hydrodynamic limit fluctuations around the macroscopic flow are the same as those given by the "averaged medium," the limit being a generalized stable Ornstein-Uhlenbeck process. The convergence proof is based on an analysis of a nonlinear integral equation with random coefficients.

Let $\{\mathbf{Z}_n\}$ be a multitype branching process in a random environment (MBPRE) which grows to infinity with positive probability for almost all environmental sequences. Under some conditions involving the first two moments of the environmental sequence, it is shown that dividing the $\{\mathbf{Z}_n\}$ components by their environment-conditioned expectations yields a sequence convergent in $L^2$ to a random vector with equal components.

Let $\{X_n\}$ be a Markov chain on a bounded set in $R^d$ with $E_x(X_1) = f_N(x) = x + \beta_N h_N(x)$, where $x_0$ is a stable fixed point of $f_N(x) = x$, and $\operatorname{Cov}_x(X_1) \approx \sigma^2(x)/N$ in various senses. Let $D$ be an open set containing $x_0$, and assume $h_N(x) \rightarrow h(x)$ uniformly in $D$ and either $\beta_N \equiv 1$ or $\beta_N \rightarrow 0, \beta_N \gg \sqrt{\log N/N}$. Then, assuming various regularity conditions and $X_0 \in D$, the time the process takes to exit from $D$ is logarithmically equivalent in probability to $e^{VN\beta_N}$, where $V > 0$ is the solution of a variational problem of Freidlin-Wentzell type $\lbrack \text{if} \beta_N \rightarrow 0 \text{and} d = 1, V = \inf\{2 \int^y_{x_0}\sigma^{-2}(u)|h(u) du|: y \in \partial D\} \rbrack$. These results apply to the Wright-Fisher model in population genetics, where $\{X_n\}$ represent gene frequencies and the average effect of forces such as selection and mutation are much stronger than effects due to finite population size.

This note proves a uniform large deviation property for the empirical process of certain Markov chains that take values in a Polish space. The proof is based on recent results for the empirical measure.

Consider two random sequences $X_1 \cdots X_n$ and $Y_1 \cdots Y_n$ of i.i.d. letters in which the probability that two distinct letters match is $p > 0$. For each value $a$ between $p$ and 1, the length of the longest contiguous matching between the two sequences, requiring only a proportion $a$ of corresponding letters to match, satisfies a strong law analogous to the Erdos-Renyi law for coin tossing. The same law applies to matching between two nonoverlapping regions within a single sequence $X_1 \cdots X_n$, and a strong law with a smaller constant applies to matching between two overlapping regions within that single sequence. The method here also works to obtain the strong law for matching between multidimensional arrays, between two Markov chains and for the situation in which a given proportion of mismatches is required.

Let $S_n = X_1 + \cdots + X_n$, where $X_1, X_2, \cdots$ are iid Banach-space-valued random variables with weak mean 0 and weak second moments. Let $K$ be the unit ball of the reproducing kernel Hilbert space associated to the covariance of $X$. The cluster set $A$ of $\{S_n/(2n \log \log n)^{1/2}\}$ is known to be a.s. either empty or have form $\alpha K$, with $0 \leq \alpha \leq 1$ determined by a series condition. To show that this series condition is a complete characterization of $A$, examples are given to show that all $\alpha \in \lbrack 0, 1)$ do occur; $A = \phi$ and $\alpha = 1$ are already known possibilities. A regularity condition is given under which $A$ must be either $\phi$ or $K$. Under stronger moment conditions, a natural necessary and sufficient condition for $A = \phi$ is given.

Let $\{X_i\}$ be i.i.d. and $S_n(s_n, r_n)$ the sum of the first $n X_i$ with the $r_n$ largest and $s_n$ smallest excluded. Assume $r_n \rightarrow \infty, s_n \rightarrow \infty, n^{-1}r_n \rightarrow 0, n^{-1}s_n \rightarrow 0.$ Necessary and sufficient conditions are obtained for the existence of $\{\delta_n\}, \{\gamma_n\}$ such that $\gamma^{-1}_n(S_n(s_n, r_n) - \delta_n)$ converges weakly to a standard normal. The set of all subsequential limit laws for these sequences is characterized and sufficient conditions are given for $X_i$ to be in the domain of partial attraction of a given law in the class. These conditions are also necessary if a unique factorization result for characteristic functions is true.

This paper establishes a functional central limit theorem for Lie groups under a mixing hypothesis. The main theorem generalizes results by Patrick Billingsley for Euclidean space and the author for the general linear group.

Let $X, X_1, X_2, \ldots$ be i.i.d. mean zero random variables. Put $S_k = X_1 + \cdots + X_k$. We prove that for every $n \geq 1, E \max_{1 \leq k \leq n} S^+_n \leq (2 - n^{-1})ES^+_n$. This result is nearly sharp, since if $P(X = 1) = P(X = -1) = \frac{1}{2},$ then $E \max{1 \leq k \leq n} S^+_k = (2 - n^{-1/2}\gamma^+_n)ES^+_n,$ where $\lim_{n \rightarrow \infty} \gamma^+_n = \sqrt{\pi/2}$.

Let $M$ be a complete Riemannian manifold and $p(t, x, y)$ the minimal heat kernel on $M$. Let $P_t$ be the associated semigroup. We say that $M$ is stochastically complete if $\int_M p(t, x, y) dy = 1$ for all $t > 0, x \in M$; we say that $M$ has the $C_0$-diffusion property (or the Feller property) if $P_tf$ vanishes at infinity for all $t > 0$ whenever $f$ is so. Let $x_0 \in M$ and let $\kappa(r)^2 \geq -\inf\{Ric(x): \rho(x, x_0) \leq r\}$ ($\rho$ is the Riemannian distance). We prove that $M$ is stochastically complete and has the $C_0$-diffusion property if $\int^\infty_c \kappa(r)^{-1} dr = \infty$ by studying the radial part of the Riemannian Brownian motion on $M$.

An urn model is defined as follows: $n$ balls are independently placed in an infinite set of urns and each ball has probability $p_k > 0$ of being assigned to the $k$th urn. We assume that $p_k \geq p_{k + 1}$ for all $k$ and that $\sum^\infty_{k = 1} p_k = 1$. A random variable $Z_n$ is defined to be the number of occupied urns after $n$ balls have been thrown. The main result is that $Z_n$, when normalized, converges in distribution to the standard normal distribution. Convergence to $N(0, 1)$ holds for all sequences $\{p_k\}$ such that $\lim_{n \rightarrow \infty} \operatorname{Var}Z_{N(n)} = \infty$, where $N(n)$ is a Poisson random variable with mean $n$. This generalizes a result of Karlin.

Let $(X, Y)$ be a pair of Hilbert-valued random variables for which the Wasserstein distance between the marginal distributions is reached. We prove that the mapping $\omega \rightarrow (X(\omega), Y(\omega))$ is increasing in a certain sense. Moreover, if $Y$ satisfies a nondegeneration condition, we can take $X = T(Y)$ with $T$ monotone in the sense of Zarantarello. We apply these results to obtain a proof of the central limit theorem (CLT) in Hilbert spaces which does not make use of the CLT for real-valued random variables.