We count the number of times a random walk exits from a square root boundary and show that the normalized counting process and the normalized random walk converge jointly in law to a "local time," whose inverse is a stable subordinator of a known index, and a Brownian motion. The study of this limit process leads to some precise sample path properties of Brownian motion. These properties improve earlier results of Dvoretsky and Kahane on the existence of small oscillations in the Brownian path.

Let $\{X_n\}$ be a stationary sequence of random variables whose marginal distribution $F$ belongs to a stable domain of attraction with index $\alpha, 0 < \alpha < 2$. Under the mixing and dependence conditions commonly used in extreme value theory for stationary sequences, nonnormal stable limits are established for the normalized partial sums. The method of proof relies heavily on a recent paper by LePage, Woodroofe, and Zinn which makes the relationship between the asymptotic behavior of extreme values and partial sums exceedingly clear. Also, an example of a process which is an instantaneous function of a stationary Gaussian process with covariance function $r_n$ behaving like $r_n \log n \rightarrow 0$ as $n \rightarrow \infty$ is shown to satisfy these conditions.

A new proof of the Hartman-Wintner law of the iterated logarithm is given. The main new ingredient is a simple exponential inequality. The same method gives a new, simpler proof of a basic result of Kuelbs on the LIL in the Banach space setting.

We identify a representation problem involving the Radon transforms of signed measures on $\mathbb{R}^d$ of finite total variation. Specifically, if $\mu$ is a pointwise translate of $v$ (i.e., if for all $\theta \in S^{d - 1}$ the projection $\mu_\theta$ is a translate of $v_\theta$), must $\mu$ be a vector translate of $v$? We obtain results in several important special cases. Relating this to limit theorems, let $X_{n1}, \cdots, X_{nk_n}$ be a u.a.n. triangular array on $\mathbb{R}^d$ and put $S_n = X_{n1} + \cdots + X_{nk_n}$. There exist vectors $v_n \in \mathbb{R}^d$ such that $\mathscr{L}(S_n - v_n) \rightarrow \gamma$ iff (I) a tail probability condition, (II) a truncated variance condition, and (III) a centering condition hold. We find that condition (III) is superfluous in that (I) and (II) always imply (III) iff the limit law $\gamma$ has the property that the only infinitely divisible laws which are pointwise translates of $\gamma$ are vector translates. Not all infinitely divisible laws have this property. We characterize those which do. A physical interpretation of the pointwise translation problem in terms of the parallel beam x-ray transform is also discussed.

This paper develops precise estimates for the potential kernel, capacities of large intervals, and the probabilities of hitting large intervals for the asymmetric Cauchy processes. These are then applied to study three problems concerning the sample paths: (i) the rate of escape of $|X_t|$ as $t \rightarrow \infty$; (ii) the sizes of the large holes in the range of the process; (iii) the asymptotic behavior of the Lebesgue measure of that part of the range of the process that is in a large interval.

Let $F_n$ be the Uniform empirical distribution function. Write $\hat F_n$ for the (least) concave majorant of $F_n$, and let $\hat f_n$ denote the corresponding density. It is shown that $n \int^1_0 (\hat f_n(t) - 1)^2 dt$ is asymptotically standard normal when centered at $\log n$ and normalized by $(3 \log n)^{1/2}$. A similar result is obtained in the 2-sample case in which $\hat f_n$ is replaced by the slope of the convex minorant of $\bar F_m = F_m \circ H^{-1}_N$.

It is shown that under appropriate conditions the $s$th cumulant of a von Mises statistic or a $U$ (or $V$) statistic is $O(n^{-s + 1}), s \geq 2$, as the sample size $n$ goes to infinity. A possible route toward the derivation of an asymptotic expansion of the characteristic function is indicated.

Rates of convergence in the central limit theorem are frequently described in terms of the uniform metric. However, statisticians often apply the central limit theorem only at symmetric pairs of isolated points, such as the 5% points of the standard normal distribution, $\pm 1.645$. In this paper we study rates of convergence on sets of the form $\{-\theta, \theta\}$, where $\theta \geq 0$. It is shown that the rate of convergence on the 5% points is the same as the rate uniformly on the whole real line, up to terms of order $n^{-1/2}$. Curiously, the rate of convergence on the 1% points $\pm 2.326$ can be faster than the rate on the whole real line.

Consider a system of particles moving on the integers with a simple exclusion interaction: each particle independently attempts to execute a simple symmetric random walk, but any jump which would carry a particle to an already occupied site is suppressed. For the system running in equilibrium, we analyze the motion of a tagged particle. This solves a problem posed in Spitzer's 1970 paper "Interaction of Markov Processes." The analogous question for systems which are not one-dimensional, nearest-neighbor, and either symmetric or one-sided remains open. A key tool is Harris's theorem on positive correlations in attractive Markov processes. Results are also obtained for the rightmost particle in the exclusion system with initial configuration $Z^-$, and for comparison systems based on the order statistics of independent motions on the line.

The fluctuations of an infinite system of unscaled branching Brownian motions in $R^d$ are shown to converge weakly under a spatial central limit normalization when the initial density of particles tends to infinity. The limit is a generalized Gaussian process $M$ which can be written as $M = M^I + M^{II}$, where $M^I$ is the fluctuation limit of a Poisson system of Brownian motions obtained by Martin-Lof, and $M^{II}$ arises from the spatial central limit normalization of the "demographic variation process" of the system. In the critical case $M^I$ and $M^{II}$ are independent and $M^{II}$ coincides with the generalized Ornstein-Uhlenbeck process found by Dawson and by Holley and Stroock as the renormalization limit of an infinite system of critical branching Brownian motions when $d \geq 3$. Generalized Langevin equations for $M, M^I$ and $M^{II}$ are given.

An old argument of Royden and Tsuji is modified to give a necessary and sufficient condition for a reversible countable state Markov chain to be transient. This Royden criterion is quite convenient and can, on occasion, be used as a substitute for the criterion of Nash-Williams [6]. The result we give here yields a very simple proof that the Nash-Williams criterion implies recurrence. The Royden criterion also yields as a trivial corollary that a recurrent reversible random walk on a state space $X$ remains recurrent when it is constrained to run on a subset $X'$ of $X$. An apparently weaker criterion for transience is also given. As an application, we discuss the transience of a random walk on a horn shaped subset of $\mathbb{Z}^d$.

Consider the number of steps needed by algorithms to locate the minimum of functions defined on the $d$-cube, where the functions are known to have no local minima except the global minimum. Regard this as a game: one player chooses a function, trying to make the number of steps needed large, while the other player chooses an algorithm, trying to make this number small. It is proved that the value of this game is approximately of order $2^{d/2}$ steps as $d \rightarrow \infty$. The key idea is that the hitting times of the random walk provide a random function for which no algorithm can locate the minimum within $2^{d(1/2 - \varepsilon)}$ steps.

Consider a random walk on $\mathbb{R}^d$ with stationary, possibly dependent increments. Let $N(V)$ count the number of visits to a bounded set $V$. We give bounds on the size of $N(t + V)$, uniformly in $t$, in terms of the behavior of $N$ in a neighborhood of the origin.

It is shown that if $F$ is an IMRL (increasing mean residual life) distribution on $\lbrack 0, \infty)$ then: $\max\{\sup_t |\bar F(t) - \bar G(t)|, \sup_t|\bar F(t) - e^{-t/\mu}|, \sup_t|\bar G(t) - e^{-t/\mu}|, \\ \sup_t |\bar G(t) - e^{-t/\mu_G}|\} = \frac{\rho}{\rho + 1} = 1 - \frac{\mu}{\mu_G}$ where $\bar F(t) = 1 - F(t), \mu = E_FX, \mu_2 = E_FX^2, G(t) = \mu^{-1} \int^t_0 \bar F(x) dx, \mu_G = E_GX = \mu_2/2\mu$, and $\rho = \mu_2/2\mu^2 - 1 = \mu_G/\mu - 1$. Thus if $F$ is IMRL and $\rho$ is small then $F$ and $G$ are approximately equal and exponentially distributed. IMRL distributions with small $\rho$ arise naturally in a class of first passage time distributions for Markov processes, as first illuminated by Keilson. The current results thus provide error bounds for exponential approximations of these distributions.

For $f$, a random single-valued mapping of an $n$-element set $X$ into itself, let $f^{-1}$ be the inverse mapping, and $f^\ast$ be such that $f^\ast(x) = \{f(x)\} \cup f^{-1}(x), x \in X$. For a given subset $A \subset X$, introduce three random variables $\xi(A) = |\hat f(A)|, \eta(A) = |\hat f^{-1}(A)|$, and $\zeta(A) = |\hat f^\ast(A)|$, where $\hat f, \hat f^{-1}, \hat f^\ast$ stand for transitive closures of $f, f^{-1}, f^\ast$. The distributions of $\xi(A)$ and $\zeta(A)$ are obtained. ($\eta(A)$ was earlier studied by J. D. Burtin.) For large $n$, the asymptotic behavior of those distributions is studied under various assumptions concerning $m = |A|$. For instance, it is shown that $\xi(A)$ is asymptotically normal with mean $(2mn)^{1/2}$ and variance $n/2$, and $(n - \zeta(A))(n/m)^{-1}$ is asymptotically $\mathscr{U}^2/2$ ($\mathscr{U}$ being the standard normal variable), provided $m \rightarrow \infty, m = o(n)$. The results are interpreted in terms of epidemic processes on random graphs introduced by I. Gertsbakh.

Comparisons are made between the expected returns using measurable and non-measurable stop rules in discrete-time stopping problems. In the independent case, a natural sufficient condition ("preservation of independence") is found for the expected return of every bounded non-measurable stopping function to be equal to that of a measurable one, and for that of every unbounded non-measurable stopping function to be arbitrarily close to that of a measurable one. For non-negative and for uniformly-bounded independent random variables, universal sharp bounds are found for the advantage of using non-measurable stopping functions over using measurable ones. Partial results for the dependent case are obtained.

A simple extension of Slepian's inequality is given which implies the original inequality and the result that positively correlated normal random variables are associated. Extensions to elliptically contoured distributions are given.