A convex set $M$ is called a simplex if there exists a subset $M_e$ of $M$ such that every $P \in M$ is the barycentre of one and only one probability measure $\mu$ concentrated on $M_e$. Elements of $M_e$ are called extreme points of $M$. To prove that a set of functions or measures is a simplex, usually the Choquet theorem on extreme points of convex sets in linear topological spaces is cited. We prove a simpler theorem which is more convenient for many applications. Instead of topological considerations, this theorem makes use of the concept of sufficient statistics.

Herein, laws of the iterated logarithm and various upper-lower class refinements are established for partial sums of stationary Gaussian random variables. These stationary Gaussian random variables are not necessarily in any sense weakly dependent. For example, if the random variables are nonnegatively correlated, then the upper half of the law of the iterated logarithm holds. Under more restrictive, but still quite general hypotheses, an upper-lower class test which classifies all monotone sequences $\{\phi(n)\}$ is established.

Let $\{W_n\}_{n=1,2,\cdots}$ denote a sequence of $k$-dimensional random vectors on a probability space $(\Omega, \mathscr{A}, P)$. Using moment-generating function techniques sufficient conditions are given for the existence of limits $\rho(A) = \lim_{n\rightarrow\infty} \lbrack P(W_n \not\in k_n A)\rbrack^{1/k_n}$ for certain subsets $A \subset R^k$, where $\{k_n\}_{n=1,2,\cdots}$ is a divergent sequence of positive real numbers. The results are multivariate analogs of well-known large deviation theorems on the real line.

The Bessel function ratios $(b/a)^\nu K_\nu(as^{\frac{1}{2}}) (a > b > 0, \nu \in R)$ and $(b/a)^\nu I_\nu(as^{\frac{1}{2}})/I_\nu(bs^{\frac{1}{2}}) (0 < a < b, \nu > -1)$ are infinitely divisible Laplace transforms in $s > 0$. These results are derived as hitting times of the Bessel diffusion process. The infinite divisibility of the $t$-distribution is deduced as a limiting result. A relationship with the von Mises-Fisher distribution is also demonstrated.

For asymmetric Cauchy processes an integral test is given for the sample function growth immediately after the process has been zero for the last time.

With any Harris-recurrent Markov chain one can associate a sequence of random times at which the chain has the same distribution, and the chain can thereby be shown to be equivalent to one having a recurrence point. This idea makes available a regeneration scheme for such chains, which is exploited in this paper to prove the ergodic theorem for semi-Markov processes, and a renewal theorem for Markov chains on a general state space.

Let $\{v_k, k \geqq 0\}$ be a discrete time Markov process with state space $E \subset (-\infty, \infty)$ and let $S$ be a proper subset of $E$. In several applications it is of interest to know the behavior of the system after a large number of steps, given that the process has not entered $S$. In this paper we show that under some mild restrictions there is a functional limit theorem for the conditioned sequence if there was one for the original sequence. As applications we obtain results for branching processes, random walks, and the M/G/1 queue which complete or extend the work of previous authors. In addition we consider the convergence of conditioned birth and death processes and obtain results which are complete except in the case that 0 is an absorbing boundary.

Conditions are given for a sequence of stochastic processes derived from row sums of an array of dependent random variables to converge to a process with stationary, independent increments or to a process with continuous paths. We also discuss when row maxima converge to an extremal process. The first result is a generalization of the classical results for independent random variables. The second result gives general conditions for convergence to processes which can be obtained from Brownian motion by a random change of time. This result is used to give a unified development of most of the martingale central limit theorems in the literature. An important aspect of our methods is that after the initial result is shown, we can avoid any further consideration of tightness.

In this paper we study extremes of non-normal stable moving average processes, i.e., of stochastic processes of the form $X(t) = \Sigma a(\lambda - t)Z(\lambda)$ or $X(t) = \int a(\lambda - t) dZ(\lambda)$, where $Z(\lambda)$ is stable with index $\alpha < 2$. The extremes are described as a marked point process, consisting of the point process of (separated) exceedances of a level together with marks associated with the points, a mark being the normalized sample path of $X(t)$ around an exceedance. It is proved that this marked point process converges in distribution as the level increases to infinity. The limiting distribution is that of a Poisson process with independent marks which have random heights but otherwise are deterministic. As a byproduct of the proof for the continuous-time case, a result on sample path continuity of stable processes is obtained.

This paper is concerned with the Hawkins random sieve which is a probabilistic analogue of the sieve of Eratosthenes. Analogues of the prime number theorem, Mertens' theorem and the Riemann hypothesis have previously been established for the Hawkins sieve. In the present paper we give a more delicate analysis using iterated logarithm results for both martingales and tail sums of martingale differences to deduce a considerably improved $\log\log$ replacement for the Riemann hypothesis result.

Stone has used Fourier analytic methods to show that the renewal function $U(x) = \sum^\infty_0 F^{n\ast}(x)$ for a random variable $X$ with distribution function $F$, finite second moment and positive mean $\lambda^{-1} = EX$, is bounded above by $\lambda x_+ + C\lambda^2EX^2$ for a universal constant $C, 1 \leqq C < 3$. This paper refines his method to prove that $C < 2.081$, and shows that within certain constraints the smallest upper bound on $C$ that the method will yield is 1.809. Various authors' work on the simpler case where $X \geqq 0$ is summarized: the best result is the earliest published one, due to Lorden, who showed that then $C = 1$.

Consider a system of noninteracting particles moving at constant speeds in a Euclidean space. It is shown that, if at two or more epochs the velocities are independent of the positions and are independent and identically distributed, then the positions must be given by a Cox process of a very special structure.

Wiener and Masani describe a procedure for relating nonlinear prediction of a univariate random process to linear prediction of an infinite-variate process which may not be a Hilbert-space-valued process but may be Banach-space-valued instead. An algorithm for computation of the linear predictor and the generating function of a Banach-space-valued stationary stochastic process is obtained under an extension of the boundedness condition of Wiener and Masani on the spectral density of the process.