The stepping stone model describes a situation in which beasts alternately migrate among an infinite array of colonies, undergo random mating within each colony, and are subject to selectively neutral mutation at the rate $u$. Assume the beasts follow a random walk $\{X_n\}$. If $u = 0$, we show that two randomly chosen beasts in the $n$th generation in any bounded set are genetically identical at a given locus with probability converging to one iff the symmetrization of $\{X_n\}$ is recurrent. In general, if either $u = 0$ or $u$ is of order $1/n$, this probability converges to its limit at the rate $C/n^{\frac{1}{2}}$ for finite variance walks in one dimension and $C/(\log n)^a$ in two, with other rates for other classes of $\{X_n\}$. More complicated rates ensure for $u \neq O(1/n)$.

Let $\mathbf{X}_1, \mathbf{X}_2, \cdots$ be a sequence of i.i.d. rv's uniformly distributed over the unit square $I^2$. Further, let $F_n$ be the empirical distribution function based on the sample $\mathbf{X}_1, \mathbf{X}_2, \cdots, \mathbf{X}_n$. A sequence $\{B_n\}$ of Brownian bridges and a Kiefer process $K$ is constructed such that $$\sup_{A\in Q} |n^{\frac{1}{2}}(F_n(A) - \lambda(A)) - B_n(A)| = O(n^{-\frac{1}{19}}) \\ \sup_{A\in Q} |n(F_n(A) - \lambda(A)) - K(A; n)| = O(n\frac{1}{2}\frac{2}{5})$$ a.s. where $F_n(A), B_n(A), K(A; n)$ are the corresponding random measures of $A, \lambda$ is the Lebesgue measure and $Q$ is the set of Borel sets of $\mathrm{I}^2$ having twice differentiable boundaries.

We prove a strong convergence theorem for Banach space valued random variables. One corollary of this result establishes necessary and sufficient conditions for the law of the iterated logarithm (LIL) in the Banach space setting. We also prove an exact generalization of the Hartman-Wintner law of the iterated logarithm provided the random variables involved take values in a real separable Hilbert space or some other Banach space with smooth norm.

Let $N$ be the positive integers. We define a class of processes indexed by $N^r \times N^r$ which we call subadditive (when $r = 1$ our definition coincides with the usual one). Under a first moment condition we prove mean convergence of $x_{0t}/|\mathbf{t}|$ as each coordinate of $\mathbf{t} \rightarrow \infty$, where $|\mathbf{t}| = t_1 t_2 \cdots t_r$. If the process is strongly subadditive (a more restrictive condition) then the same first moment condition gives a.s. sectorial convergence. We conjecture (and verify in several cases) that an $L(\log L)^{r-1}$ integrability condition is sufficient to give unrestricted a.s. convergence.

Let $p(x, y)$ be an irreducible symmetric transition function for a Markov chain on a countable set $S$. Let $\eta_t$ be the infinite particle system on $S$ with simple exclusion interaction modified to allow the spontaneous creation and destruction of particles in the system. A complete characterization of the invariant probability measures for this system is obtained in the case where the exponential rates of creation and destruction are independent of the configuration of the system. Furthermore, if $\mathscr{M}$ is the set of probability measures on the state space of $\eta_t$ and $S(t)$ is the semigroup on $\mathscr{M}$ determined by $$S(t)\mu(A) = \int P^\eta\lbrack\eta_t \in A\rbrack d\mu(\eta)$$ theorems concerning the weak convergence of $S(t)\mu$ to the invariant measures of $\eta_t$ are proved.

In this paper we give examples of the sets of recurrent points (or accumulation points) of random walks in $R^d$ normed by nice sequences of constants. These examples, interesting in their own right, give rise to some very interesting conjectures concerning the general structure of such sets. Of particular interest are the recurrent points of the ordinary averages or sample means. It turns out that any closed subset of $R^d$ can be the finite points of recurrence of a sequence of averages: $(X_1 + \cdots + X_n)/n, X_i$ i.i.d. random vectors. This seems to be a property not shared by most other normalizing sequences. We also give some results on rates of escape of random walks in a domain of attraction. In looking for rates of escape we are looking for normalizing constants which give rise to no finite recurrent points of the normalized walk.

Let $\{X(t), -\infty < t < \infty\}$ be a stochastic process which is stationary in the wide sense with spectral representation $X(t) = \int^\infty_{-\infty} e^{it\lambda} d\xi(\lambda)$, where the $\xi$ process is centered and has independent increments with $E\xi(\lambda) \equiv 0, E|\xi(\lambda)|^2 < \infty$. It is shown that under weak conditions $$P - \lim_{T\rightarrow\infty} \frac{1}{2T} \int^T_{-T} |X(t)|^2 dt$$ exists and is equal to $\sigma^2 + \sum J_t^2 + \sum \xi_n^2$, where $\sigma^2$ is equal to the variance of the Gaussian component of the continuous part of the $\xi$ process, $\sum J_t^2$ is the sum of the squares of the jumps of the Gaussian component of the $\xi$ process, and $\xi_N = \xi(\lambda_N + 0) - \xi(\lambda_N - 0)$, where $\{\lambda_N\}$ are the fixed discontinuities of the $\xi$ process.

Let $\lbrack X_{ij}: i > 0, j > 0 \rbrack$ be a double sequence of i.i.d. random variables taking values in the $d$-dimensional lattice $E_d$. Also let $S_{mn} = \sum^m_{i=1} \sum^n_{j=1} X_{ij}$. Then the range of random walk $\lbrack S_{mn}: m > 0, n > 0 \rbrack$ up to time $(m, n)$, denoted by $R_{mn}$, is the cardinality of the set $\lbrack S_{pq}: 0 < p \leqq m, 0 < q \leqq n \rbrack$, i.e., the number of distinct points visited by the random walk up to time $(m,n)$. In this paper a strong law for $R_{mn}$, when $d \geqq 3$, has been established. Namely, it has been proved that $\lim R_{mn}/ER_{mn} = 1$ a.s. as either $(m, n)$ or $m (n)$ tends to infinity.

Let $X(t, \omega, a \leqq t \leqq b, \omega \in \Omega$ be a real continuous stationary Gaussian process with mean 0 and covariance $R$. We prove that there exist analytic functions $f_n$ defined on $\lbrack a, b\rbrack$ and independent random variables $X_nN(0, 1), n = 0,1,2, \cdots$, such that the series $\sum^\infty_{n=0} f_n(t)X_n$ converges uniformly to $X(t)$ with probability 1. Among other applications of this representation theorem, we show that if the second spectral moment is infinite and $\int^\delta_0 (R(0) - R(t))^{-\frac{1}{2}} dt < \infty$ for some $0 < \delta \leqq b - a$, then for any given $u\in\mathbb{R}, P\{\omega\mid X_\omega^{-1}(u)$ is infinite$\} > 0$.

Coupling methods are used to obtain a structure theorem for the atomic decomposition of the tail $\sigma$-algebra of an arbitrary nonhomogeneous Markov chain. Various related results are also derived by coupling.

An improved result is presented, showing that if the finite-dimensional distributions of a sequence of martingales converge, and if for each time $t$ the variables are uniformly integrable, then weak convergence follows (in either $C$ or $D$) provided the limiting process satisfies a certain condition; this condition is satisfied by the Wiener process.

Let $\{S_n\}, n = 1,2, \cdots$ denote the partial sums of a sequence of independent, identically distributed nonnegative random variables with common distribution function $F$ having finite mean $\mu$, and let $H(t) = \sum^\infty_{n=1} P(S_n \leqq t)$. Further, let $F$ be nonarithmetic. It is shown in this paper that as $t \rightarrow \infty H(t) - t/\mu$ is regularly varying if and only if $F$ belongs to the domain of attraction of a stable law with exponent $\alpha, 1 < \alpha \leqq 2$.

Consider the class of probability distributions on the unit square with uniform marginal distributions. The class of associated regression functions is characterized by integral inequalities or, equivalently, as forming the closed convex hull of 1-1, measure-preserving mappings of the unit interval onto itself.