It is well known that the voter model on $\mathbb{Z}^d$ under initial product measure will converge weakly to a point process as $t \rightarrow \infty$; if $d \geqslant 3$, convergence will be to a nontrivial point process; if $d = 1,2$, convergence will be to a linear combination of trivial point processes. Therefore, for $d = 1,2$, the cluster size of a particular state around any fixed point tends to become arbitrarily large as $t \rightarrow \infty$. Here we examine the rate of growth for $d = 1$ of this clustering for the nearest neighbor voter model and the related problem of interparticle distance for nearest neighbor coalescing random walks and annihilating random walks. We show that under spatial renormalization $t^{\frac{1}{2}}$ these cluster sizes/interparticle distances in each case approach a nondegenerate distribution. We examine these distributions and obtain numerical estimates for these and related problems.

Let $f$ be a continuous function from the unit interval to itself and let $X_0, X_1, \cdots$ be the successive proportions of red balls in an urn to which at the $n$th stage a red ball is added with probability $f(X_n)$ and a black ball with probability $1 - f(X_n)$. Then $X_n$ converges almost surely to a random variable $X$ with support contained in the set $C = \{p: f(p) = p\}$. If, in addition, $0 < f(p) < 1$ for all $p$, then, for each $r$ in $C, P\lbrack X = r\rbrack > 0(=0)$ when $f'(r) < 1(> 1)$. These results are extended to more general functions $f$.

Renewal processes with increasing mean residual life and decreasing failure rate interarrival time distributions are investigated. Various two-sided bounds are obtained for $M(t)$, the expected number of renewals in $\lbrack 0, t\rbrack$. It is shown that if the interarrival time distribution has increasing mean residual life with mean $\mu$, then the expected forward recurrence time is increasing in $t \geqslant 0$, as is $M(t) - t/\mu$. If the interarrival time distribution has decreasing failure rate then $M(t)$ is concave, and the forward and backward recurrence time distributions are stochastically increasing in $t \geqslant 0$.

We consider the approximation of an Ito integral $\int^t_0 \phi(s)dB(s)$ by a sequence of integrals $\int^t_0 \phi_n(s)dB(s)$ of simpler integrands. It is proved that if, for a sequence $\{\psi_n\}$ of adapted integrands, $\sup_{0\leqslant t\leqslant 1}|\int^t_0 \psi_n ds| \rightarrow_p 0$ and $\int^t_0 \psi^2_n ds\rightarrow_p\tau(t)$, for some continuous stochastic process $\{\tau(t); t \in\lbrack 0, 1\rbrack\}$, then $\int_0^{(\cdot)}\psi_ndB \rightarrow_d W \circ \tau$ in $C(0, 1)$, where $W$ is a Brownian motion independent of $\tau$. Further, if one is only interested in the limit distribution of functionals like $\int^1_0\psi_n dB$ or $\sup_{0\leqslant t\leqslant 1}| \int^t_0\psi_n dB|$, then in the second condition it is enough to require that $\int^1_0\psi^2_nds \rightarrow_p\tau(1)$. The convergence is stable in the sense of Renyi, and from this follow results on the fluctuations of the sample paths of the integrals. As an example we consider the case $\phi(t) = f(B(t), t)$ and $\phi_n(t) = \sum^n_{i=1} f(B(i/n), i/n)I(i/n \leqslant t < (i + 1)/n)$. Denoting the approximation error $\int^t_0(\phi - \phi_n)dB$ by $d_n(t)$, it follows from the above results that if $f$ is smooth enough then $n^{\frac{1}{2}}d_n \rightarrow_d W \circ \tau$, with $\tau(t) = 2^{-1} \int^t_0 f_1(B(s), s)^2ds$ where $f_1(x, t) = \frac{\partial f(x, t)}{\partial x}$. Similar results are obtained for approximations of the Stratonovich integral and for higher order approximations.

This paper establishes an almost sure limit for the operator norm of rectangular random matrices: Suppose $\{v_{ij}\}i = 1,2, \cdots, j = 1,2, \cdots$ are zero mean i.i.d. random variables satisfying the moment condition $E|\nu_{11}|^n \leqslant n^{\alpha n}$ for all $n \geqslant 2$, and some $\alpha$. Let $\sigma^2 = Ev^2_{11}$ and let $V_{pn}$ be the $p \times n$ matrix $\{v_{ij}\}_{1\leqslant i\leqslant p; 1\leqslant j\leqslant n}$. If $p_n$ is a sequence of integers such that $p_n/n \rightarrow y$ as $n \rightarrow \infty$, for some $0 < y < \infty$, then $1/n|V_{p_nn}V^T_{p_nn}| \rightarrow (1 + y^{\frac{1}{2}})^2\sigma^2$ almost surely, where $|A|$ denotes the operator ("induced") norm of $A$. Since $1/n|V_{p_nn}V^T_{p_nn}|$ is the maximum eigenvalue of $1/nV_{p_nn}V^T_{p_nn}$, the result relates to studies on the spectrum of symmetric random matrices.

Let $S_n$ be a sequence of partial sums of mean zero purely $d$-dimensional i.i.d. random vectors. Necessary and sufficient conditions are given for the existence of matrices $A_n$ such that the transform of $S_n$ by $A_n$ is asymptotically multivariate normal with identity covariance matrix. This is more general than previous $d$-dimensional results. Examples are given to illustrate the need for the present approach. The matrices $A_n$ take a particularly simple form because of a degree of uncorrelatedness between certain pairs of 1-dimensional random variables obtained by projection.

Speeds of convergence to normality and asymptotic expansions for sums of independent random vectors in $\mathbb{R}^k, k \geqslant 1$ are investigated using the method of operators. Existing results are improved and some new results obtained. In particular, asymptotic expansions for smooth functions are derived.

For a set of i.i.d. random variables indexed by $Z^d_+, d \geqslant 1$, the positive integer $d$-dimensional lattice points, convergence rates for moderate deviations are derived, i.e., the rate of convergence to zero of, for example, certain tail probabilities of the partial sums, are determined. As an application we obtain results on the integrability of last exit times (in a certain sense) and the number of boundary crossings of the partial sums.

Let $\{S_n\}$ be the partial sums of a sequence of independent random variables and let $\{a_n\}$ be a nondecreasing, divergent real sequence. Necessary and sufficient conditions for $\lim \sup_{n\rightarrow\infty}S_n/a_n < \infty$ a.s. are given under mild conditions on $\{S_n\}$; these conditions do not involve the existence of any moments. These results are employed to widen the scope of the law of the iterated logarithm.

In this paper the analogue in infinite dimensions of the Erdos-Dvoretzky rate of escape test for finite dimensional Brownian motion is proved. Some examples are constructed which exhibit the essential differences between the finite and infinite dimensional cases and which suggest several conjectures and problems.

Vitali conditions $V, V', V_p, 1 \leqslant p < \infty$, on $\sigma$-algebras indexed by a directed set, are shown to hold if and only if the maximal inequality \begin{equation*}\tag{1} P(\text{essential} \lim \sup X_t \geqslant \alpha) \leqslant K \lim \sup_{T^\ast}E(X_\tau)/\alpha\end{equation*} holds for all adapted positive processes $(X_t)$, and all positive numbers $\alpha$. Here $K$ is a constant which may be taken equal to 1, and $T^\ast$ is the appropriate directed set of stopping times: for $V, T^\ast$ is the set of simple stopping times; for $V', T^\ast$ is the set of simple ordered stopping times; for $V_p, T^\ast$ is the set of multivalued stopping times with overlap going to zero in $L_p$. The inequality (1) is true whatever be the $\sigma$-algebras, provided that essential $\lim \sup$ is replaced by stochastic $\lim \sup$.

Let $S_n$ denote the $n$th partial sum of i.i.d. nonconstant mean zero random variables. Given an approximation $K(n)$ of $E|S_n|$, tight bounds are obtained for the ratio $E|S_n|/K(n)$. These bounds are best possible as $n$ tends to infinity. Implications of this result relate to the law of the iterated logarithm for mean zero variables, Chebyshev's inequality and Markov's inequality. Asymptotically exact lower-bounds are obtained for expectations of functions of row-sums of triangular arrays of independent but not necessarily identically distributed random variables. Expectations of "Poissonized random sums" are also treated.

For each value of $\beta, 0 < \beta < 2$, the integral $$\int^\infty_{-\infty} \{1 - \exp(-x^{-2}\sin^2tx)\}|t|^{-1-\beta}dt$$ decreases monotonically as a function of $x, x > 0$. This result is useful in approximating the absolute $\beta$th moment of the sum of zero mean i.i.d. random variables.

For pairs of two-state Markov processes satisfying certain conditions, the $\bar{d}$-distance is shown to be partition distance.

The classical almost sure convergence of the normed supercritical Galton-Watson branching process with finite mean is obtained by a new method which does not involve probability generating functions.

We prove the finiteness of certain exponential moments of Banach space valued random series and triangular arrays. We also prove integrability results for Poisson measures on Banach spaces.

The set of limit points associated with a generalized iterated logarithm law for sums of asymmetric i.i.d. random variables is shown to be the entire closed interval bounded by the $\lim \inf$ and the $\lim \sup$.

Let $\{X_n, n \geqslant 1\}$ be a discrete-parameter stationary Gaussian process with $E(X_i) = 0, E(X^2_i) = 1$ for all $i$ and $E(X_iX_{i+n}) = r(n)$. Let $M_n =$ maximum$(X_1, X_2 \cdots X_n)$. Under the condition that either $(\log n)^{1+\gamma}r(n) = O(1)$ as $n \rightarrow \infty$, for some $\gamma > 0$ or $\sum^\infty_{j=1}r^2(j) < \infty$, the set of all almost sure limit points of the vector sequence $$\big\{\frac{M_{1,n} - b_n}{a_n}, \frac{M_{2,n} - b_n}{a_n}, \cdots \frac{M_{p,n} - b_n}{a_n}\big\}$$ is obtained, where $(M_{j,n}), j = 1,2 \cdots p$ are independent copies of $(M_n); a_n = (\log\log n)(2 \log n)^{-\frac{1}{2}}$ and $b_n = (2 \log n)^{\frac{1}{2}}$.

We consider an infinitely divisible measure $\mu$ on a locally compact Abelian group. If $\mu \ll \lambda$ (Haar measure), and if the semigroup generated by the support of the corresponding Levy measure $\nu$ is the closure of an angular semigroup, then $\mu \sim \lambda$ over the support of $\mu$. In particular, if $\int|\chi(x) - 1|\nu(dx) < \infty$, for all characters $\chi$, or if $\nu \ll \lambda$ then $\mu \ll \lambda$ implies $\mu \sim \lambda$ over the support of $\mu$.