We examine the cluster set of $S_n/a_n$ for Banach space valued random variables, and investigate the relationship between the central limit theorem and the law of the iterated logarithm in this setting. In the case of Hilbert space valued random variables, necessary and sufficient conditions are given for the law of the iterated logarithm. Some interesting examples are also included. We then apply our results to weighted empiricals both in the supremum norm and the $L^2\lbrack 0, 1\rbrack$ norm.

Random Walk in a Random Environment, a process where a randomly chosen transition probability is assigned to each point, has been exhaustively studied in the one dimensional nearest neighbor case but is much more difficult in any more general case. This paper develops crude techniques for handling transience results and zero-one laws in a generalized set up.

A random time $R$ is called a regular birth time for a Markov Process if (i) the $R$-past and $R$-future are conditionally independent with respect to $X(R)$ and (ii) the post-$R$ process evolves as a Markov process, perhaps with different probability laws. In this paper we characterize each regular birth time in terms of an earlier, coterminal time $L$. It is shown (Theorem 4.2) that to the post-$L$ process $R$ appears as an optional time, perhaps with dependency on pre-$L$ information and on a certain invariant set.

By a backward time is meant a random time which only depends on the future, in the same sense as a stopping time only depends on the past. We show that backward times taking values in a regenerative set $M$ split $M$ into conditionally independent subsets. The conditional distributions of the past may further be identified with the Palm distributions $P_t$ with respect to the local time random measure $\xi$ of $M$ both a.e. $E\xi$ and wherever $\{P_t\}$ has a continuous version. Continuity of $\{P_t\}$ occurs essentially where $E\xi$ has a continuous density, and the latter continuity set may be described rather precisely in terms of the growth rate and regularity properties of the Levy measure of $M$.

Let $B(t)$ be a Brownian motion with local time $s(t, x)$. Paul Levy showed that for each $x, s(t, x)$ is a.s. equal to the limit as $\delta$ approaches zero of $\delta^{1/2}$ times the number of excursions from $x$, exceeding $\delta$ in length, that are completed by $B$ up to time $t$. The aim of the present paper is to show that the exceptional null sets, which may depend on $x$, can be combined into a single null set off which the above convergence is uniform in $x$. The proof uses nonstandard analysis to construct a simple combinatorial representation for the local time of a Brownian motion constructed by R. M. Anderson.

Consider a sample from a $p$-dimensional exponential family and a random vector whose distribution is the same as that of the sample given a sufficient statistic. General limit theorems for functions of sum-type for such random vectors are obtained; simple sums, linear combinations, $m$-dependent sums, and $U$-statistics are considered. The results are illustrated by some examples.

We prove that certain jump summation processes converge in distribution for the uniform topology to the Brownian sheet, while smoothed summation processes converge for various Lipschitz topologies. These results follow after a careful study of abstract, generalized Lipschitz spaces. Along the way we affirm a conjecture about smoothness and continuity of processes defined on $\lbrack 0, 1\rbrack^d$.

In this paper we define a generalization of Edgeworth expansions for the expectation of functions of normalized sums of i.i.d. Banach space valued random vectors. These expansions are valid up to $0(n^{-(s - 2)/2})$ for functions with $3(s - 2)$ bounded Frechet derivatives and random vectors with finite $s^{th}$ absolute moment.

Let $X_1, X_2, \cdots$ be a sequence of independent uniformly distributed random variables on $\lbrack 0, 1\rbrack$, and let $K_n$ be the $k$th largest spacing induced by the order statistics of $X_1, \cdots, X_{n - 1}$. We show that $\lim \sup(nK_n - \log n)/2 \log_2n = 1/k \quad\text{almost surely},$ and $\lim \inf(nK_n - \log n + \log_3n) = c \quad\text{almost surely},$ where $-\log 2 \leq c \leq 0$, and $\log_j$ is the $j$ times iterated logarithm.

Let $F$ belong to the domain of attraction of a stable law with parameters $\alpha$ and $p$. Let $X_1, X_2, \cdots$ be a sample from $F$. Put $|\tilde X_1| \leq |\tilde X_2| \leq \cdots \leq |\tilde X_n|$. We consider the asymptotic properties as $n \rightarrow \infty$ (and $k \rightarrow \infty$) of the ratio of order statistics $(\tilde X_1 + \cdots + \tilde X_{n - k})/|\tilde X_{n - k + 1}|$.

Let $G_n$ be the empirical distribution based on $n$ independent uniform random variables. Criteria for bounds on the supremum of weighted discrepancies between $G_n(u)$ and $u$ of the form: $|w_\nu(u) D_n(u)|$, where $D_n(u) = G_n(u) - u, w_\nu(u) = (u(1 - u))^{-1 + \nu}$ and $0 \leq \nu \leq 1$, are derived. Also an inequality closely related to an equality due to Daniels (1945) is given.

The purpose of this note is to demonstrate survival for the long-range contact process. This process was introduced by Spitzer [7]; it possesses the same basic evolutionary rule as does the contact process on the integers, except that particles may appear at large distances from already extant particles instead of just as immediate neighbors. We consider here two variants of this model, and show that in both cases the system will survive if (i) it commences from a reasonably dense initial state and (ii) the birth rate for particles is moderately greater than the corresponding death rate. The methodology consists primarily of an energy argument, which provides a lower bound for the particle density of the system.

Define $Z(t)$ to be the forward recurrence time at $t$ for a renewal process with interarrival time distribution, $F$, which is assumed to be IMRL (increasing mean residual life). It is shown that $E\phi(Z(t))$ is increasing in $t \geq 0$ for all increasing convex $\phi$. An example demonstrates that $Z(t)$ is not necessarily stochastically increasing nor is the renewal function necessarily concave. Both of these properties are known to hold for $F$ DFR (decreasing failure rate).

Suppose that shocks hit a device in accordance with a nonhomogeneous Poisson process with intensity function $\lambda(t)$. The $i^{th}$ shock has a value $X_i$ attached to it. The $X_i$ are assumed to be independent and identically distributed positive random variables, and are also assumed independent of the counting process of shocks. Let $D(x_1, \ldots, x_n, \underline{0}) \equiv D(x_1, \ldots, x_n, 0, 0, 0, \ldots)$ denote the total damage when $n$ shocks having values $x_1, \ldots, x_n$ have occurred. It has previously been shown that the first time that $D$ exceeds a critical threshold value is an increasing failure rate average random variable whenever (i) $\int^t_0 \lambda(s) ds/t$ is nondecreasing in $t$ and (ii) $D(\underline{x}) = \sum x_i$. We extend this result to the case where $D(\underline{x})$ is a symmetric, nondecreasing function. The extension is obtained by making use of a recent closure result for increasing failure rate average stochastic processes.

An irreducible stochastic matrix may be constructed by partitioning a line of unit length into a finite number of intervals, shifting the line to the right $(\mod 1)$ by a small amount, and defining transition probabilities in terms of the overlaps among the intervals before and after the shift. It is proved that every $2 \times 2$ irreducible stochastic matrix arises from this construction. Does every $n \times n$ irreducible stochastic matrix arise this way?

It is shown that every family of mutually singular measures in a conditional probability distribution is countable or else there is a perfect set of measures which form a strongly orthogonal family. Theorem: Let $X$ and $Y$ be complete separable metric spaces and $\mu$ a conditional probability distribution on $X \times \mathscr{B}(Y)$. Then either (1) there is a nonempty compact perfect subset $P$ of $X$ and a Borel subset $D$ of $X \times Y$ so that if $x$ and $y$ are distinct elements of $P$, then $\mu(x, D_x) = 1, \mu(y, D_x) = 0$, and $D_x \cap D_y = \phi$ or else (2) if $K$ is a subset of $X$ so that $\{\mu(x, \cdot):x \in K\}$ is a pairwise orthogonal family, then $K$ is countable.

It is pointed out that an example due to J. Pachl shows the nonexistence of regular conditional probabilities in Doob's sense when the underlying space is a perfect probability space. This answers a question of Sazonov.