Let $S_n = \sum^n_1 X_i$ be a random walk. A point $b$ is called a strong limit point of $n^{-\alpha}S_n$ if there exists a nonrandom sequence $n_k\rightarrow\infty$ such that $n_k^{-\alpha}S_{n_k}\rightarrow b$ w.p. 1. The possible structures for the set of strong limit points of $n^{-\alpha}S_n$ are determined. We also give a sufficient condition for $n^{-1}S_n$ to be dense in $\mathbb{R}$. In particular $n^{-1}S_n$ is dense in $\mathbb{R}$ when $E|X_1| = \infty$ and $n^{-1}S_n$ has a finite strong limit point.

Random stirring of the real line $R_1$ is defined. This notion is derived from a generalization of the nearest-neighbor simple exclusion model on the one-dimensional lattices discussed by Spitzer and by Harris. Under the random stirring, the motion of an infinite particle system is Markovian and has a Poisson process as an invariant probability measure. An ergodic theorem is established concerning the convergence of a system to a Poisson process.

In $R^n$, Brownian diffusion leads to the normal or Gaussian distribution. On the sphere $S^n$, diffusion does not lead to the Fisher distribution which often plays the role of the normal distribution on $S^n$. On the circle $(S^1)$ and sphere $(S^2)$, they are known to be numerically close. It is shown that there exists a random stopping time for the diffusion which leads to the Fisher distribution. This follows from the fact, proved here, that the modified Bessel function $I_v(x)$ is a completely monotone function of $v^2$ (for fixed $x > 0$). More generally, we study the class of distributions on $S^n$ which can be represented as mixtures of diffusions. The stopping time distribution is characterized, but not given in computable form. Also, three new distribution functions involving Bessel functions are presented.

Let $\{\xi_k: k \geqq 1\}$ be a sequence of independent, identically distributed random variables with $E\{\xi_1\} = 0$ and $E\{\xi_1^2\} = \sigma^2, 0 < \sigma^2 < \infty$. Form the random walk $\{S_n: n \geqq 0\}$ by setting $S_0 = 0, S_n = \xi_1 + \cdots + \xi_n, n \geqq 1$. Let $T$ denote the hitting time of the set $(-\infty, 0\rbrack$ by the random walk. The main result in this paper is a functional central limit theorem for the random functions $S_{\lbrack nt \rbrack}/\sigma n^{\frac{1}{2}}, 0 \leqq t \leqq 1$, conditional on $T > n$. The limit process, $W^+$, is identified in terms of standard Brownian motion. Similar results are obtained for random partial sums and renewal processes. Finally, in the case where $E\{\xi_1\} = \mu > 0$, it is shown that the conditional (on $T > n$) and unconditional weak limit for $(S_{\lbrack nt\rbrack} - \mu nt)/\sigma n^{\frac{1}{2}}$ is the same, namely, Brownian motion.

Central limit theorems are proved for martingales and near-martingales without the existence of moments or the full Lindeberg condition. These theorems are extended to invariance principles with a discussion of both random and nonrandom norming.

Let $\{X_k\}^\infty_{k = 1}$ be a sequence of independent centered random variables with values in $C(S)$ (i.e., sample continuous processes in $S$), $(S,d)$ being a compact metric space. This sequence is said to satisfy the central limit theorem if there exists a sample continuous Gaussian process on $S, Z$, such that $\mathscr{L}(\sum^n_{k = 1}X_k/n^{\frac{1}{2}})\rightarrow_{w^\ast} \mathscr{L}(Z)$ in $C'(C(S))$. In this paper some sufficient conditions are given for the central limit theorem to hold for $\{X_k\}^\infty_{k = 1}$; these conditions are on the modulus of continuity of the processes $X_k$ and they are expressed in terms of the metric entropy of distances associated to $\{X_k\}$. Then, in order to give some insight on these theorems, several results on the central limit theorem for particular processes (random Fourier and Taylor series, as well as more general processes on [0, 1]) are deduced.

Necessary and sufficient conditions for the weak convergence in stronger than the usual metrics of empirical, inverse empirical and partial sum processes are obtained. The results are related to work of Chibisov and of Pyke and Shorack.

The paper examines alternative postulates for subadditive processes, especially the ergodic theory thereof. It introduces superconvolutive sequences of distributions and proves limit laws for these, which generalize the weak law of large numbers, Chernoff's theorem, and Kesten's lemma. It discusses eigenshift and eigendistribution theory and concave recurrence relations in the convolutive semigroup, illustrating sundry conjectures with computer studies. It deals with applications of the theory to the first-death problem in branching processes, Bethe approximation of first-passage percolation, self-avoiding walks, maximal solutions of the generalized subconvolutive inequality, rates of convergence of a subadditive process, multidimensional subadditive processes in physics including the dimer problem and the overlapping-sphere model of liquid-vapor equilibrium, and Ulam's problem on the longest monotone subsequence of a random permutation.

W. Feller's decomposition into a road map and speed measure are adapted to symmetric Markov chains with instantaneous states. The road map is a family of matrices defined by time changing the process with measures concentrated on finite sets.

Let $X(t)$ be a stationary Gaussian process with continuous sample paths. The behavior of $|X(t)|$ as $t \rightarrow \infty$ is considered. In particular, conditions on the spectrum of the process are given which determine whether $\lim \sup_{t\rightarrow\infty}|X(t)|/(\log t)^{\frac{1}{2}} = \operatorname{Const.} > 0$. These conditions are complete except when the spectrum of the process is continuous-singular. The main concern of this paper is to study the asymptotic behavior of some specific examples of $X(t)$ with continuous-singular spectra. Many examples are given showing the asymptotic behavior of stationary Gaussian processes with discrete spectra and their indefinite integrals.

A triumvirate of sufficient conditions is given for unbounded, independent random variables to obey the Law of the Iterated Logarithm (LIL). As special cases, new results for weighted i.i.d. random variables and the Hartman-Wintner theorem are obtained. Necessity of finite variance for the two-sided LIL is shown to carry over for a large class of weighted i.i.d. random variables and the Marcinkiewicz-Zygmund example is generalized, simplified and clarified.

For series of independent random processes in the space $D\lbrack 0, 1 \rbrack$ endowed with the Skorohod topology, convergence in distribution is shown to imply almost sure convergence. Under mild conditions, such as e.g. when the limiting process has no jumps of fixed size and location, the latter convergence is uniform. As an application, we discuss a representation by Ferguson and Klass of processes with independent increments.

Let $f(t_1, \cdots, t_k)$ be the probability density function of a vector $(Y_1, \cdots, Y_k)$ of nonnegative random variables. Let the multivariate failure rate (M.F.R.) $r(t_1, \cdots, t_k)$ be defined by the ratio $f(t_1, \cdots, t_k)/P(Y_i > t_i, i = 1, 2, \cdots, k)$, for $t_i \geqq 0, i = 1, \cdots, k$. It is shown that $r(t_1, \cdots, t_k)$ is constant if and only if the distribution of $(Y_1, \cdots, Y_k)$ is a mixture of exponential distributions. Analogous results hold for the nonnegative integer valued random vector with mixture being of geometric distributions.

It is well known that the sample $p$-quantile is asymptotically normally distributed. This paper presents an estimate for the accuracy of the normal approximation. The special intention of this paper is to obtain favorable bounds for moderate samples.

It is shown that a locally compact metrizable group $G$ is compact if and only if every random walk on $G$ is recurrent.