Some theorems on first-order asymptotic behavior of probabilities of large deviations of empirical probability measures are proved. These theorems extend previous results due to Borovkov, Hoadley and Stone. A multivariate analogue of Chernoff's theorem and a large deviation result for trimmed means are obtained as particular applications of the general theory.

Let $X_1, X_2, \cdots$ be a sequence of i.i.d. random vectors taking values in a space $V$, let $\bar{X}_n = (X_1 + \cdots + X_n)/n$, and for $J \subset V$ let $a_n(J) = n^{-1} \log P(\bar{X}_n \in J)$. A powerful theory concerning the existence and value of $\lim_{n\rightarrow\infty} a_n(J)$ has been developed by Lanford for the case when $V$ is finite-dimensional and $X_1$ is bounded. The present paper is both an exposition of Lanford's theory and an extension of it to the general case. A number of examples are considered; these include the cases when $X_1$ is a Brownian motion or Brownian bridge on the real line, and the case when $\bar{X}_n$ is the empirical distribution function based on the first $n$ values in an i.i.d. sequence of random variables (the Sanov problem).

We consider a sequence of Markov chains occurring in population genetics (viz., the so-called multiallelic Wright-Fisher models) that converges weakly to a multidimensional diffusion process. Certain absorption times, which arise naturally in connection with the genetic models, are shown to also converge weakly. This extends a result of Guess. Corollaries include convergence of moments of absorption times and convergence of absorption probabilities. The latter results are used implicitly in population genetics.

Let $\{X_n; n = 0, 1, \cdots\}$ be a $\phi$-recurrent Markov chain on a general measurable state space $(S, \mathscr{F})$ with transition probabilities $P(x, A), x \in S, A \in \mathscr{F}$. The convergence of the ratio $\lambda P^{n+m}f / \mu P^ng$ (as $n \rightarrow \infty$), where $\lambda$ and $\mu$ are nonnegative measures on $(S, \mathscr{F})$ and $f$ and $g$ are nonnegative measurable functions on $S$, is studied. We show that the ratio converges, provided that $\lambda, \mu, f$ and $g$ are in a certain sense "small," and provided that for an embedded renewal sequence $\{u(n)\}$ the limit $\lim u(n + 1)/u(n)$ exists.

Place $n$ arcs, each of length $a_n$, uniformly at random on the circumference of a circle, choosing the arc length sequence $a_n$ so that the probability of completely covering the circle remains constant. We obtain the limiting distribution of the uncovered proportion of the circle. We show that this distribution has a natural interpretation as a noncentral chi-square distribution with zero degrees of freedom by expressing it as a Poisson mixture of mass at zero with central chi-square deviates having even degrees of freedom. We also treat the case of proportionately smaller arcs and obtain a limiting normal distribution. Potential applications include immunology, genetics, and time series analysis.

Necessary and sufficient conditions are given for asymptotic independence in the multivariate central limit theorem. If $\{X_n\}$ is a sequence of independent, identically distributed random variables whose common distribution is symmetric, and if the distribution of $X^2_1$ is in the domain of attraction of a stable distribution of characteristic exponent $\alpha$, then $\bar{X}$ and $s^2$ are asymptotically independent if and only if $1 \leqslant \alpha \leqslant 2$. If the components of a multivariate infinitely divisible distribution are pairwise independent, then they are independent.

Let $X_1, X_2, \cdots$ be i.i.d. real-valued random variables with $EX_1 = 0, EX^2_1 < \infty$, and $S_n = X_1 + \cdots + X_n, n = 1, 2, \cdots$. For a chosen positive integer $m$ and real $c > 0$ the exit time $N_c$ is the least integer $n \geqslant m$ such that $f(n) - cg(n) < S_n < f(n) + cg(n)$ is violated, where the functions $f$ and $g (0 < g\uparrow\infty)$ are both defined for all real $x \geqslant m$. Under certain conditions on $f$ and $g$, a function $\psi$ (unique up to an asymptotic equivalence), satisfying $\psi(x) / x \rightarrow 0$ as $x\rightarrow\infty$, is constructed on $\lbrack m, \infty)$ such that $\psi(N_c)$ is exactly exponentially bounded. This result generalizes earlier theorems of Breiman; Chow, Robbins, and Teicher; Gundy and Siegmund; Brown; and Lai. A consequence is that $N_c$ itself is not exponentially bounded. In a multivariate generalization the $X$'s take their values in $R^d$ and $N_c$ is the first exit time of $L_n$ from $(- l(c), l(c))$, where $L_n = n\Phi(S_n/n) - h(n)$, and certain conditions are imposed on $\Phi$ and $h$. Here $\psi(x) = \int^x_mh(t)t^{-1}dt$. The results are applied to show, both in the sequential $F$-test and in the Savage-Sethuraman sequential rank-order test, that for certain distributions of the $X$'s the stopping time is not exponentially bounded.

A multiplicative stochastic measure diffusion process in $R^d$ is the continuous analogue of an infinite particle branching Markov process in which the particles move in $R^d$ according to a symmetric stable process of index $\alpha 0 < \alpha \leqslant 2$. The main result of this paper is that there is a random carrying set whose Hausdorff dimension is almost surely less than or equal to $\alpha$. As a corollary it follows that the corresponding random measure is singular for $d > \alpha$. The latter result is also proved by a different approach in the case $d = \alpha$.

We seek conditions for the exchangeable $\sigma$-field of an independent non-identically-distributed sequence of random variables to be trivial. A simple necessary condition is given, and this condition is shown to be sufficient when the range space is finite. In the case of a general range space, a stronger condition is shown to be sufficient.

A new mixing condition is proposed for the study of stationary Gaussian processes on $R^1$. If the covariance function of the process is $r$, assume $$\text{Lebesgue measure}\big\{t\mid 0 \leqslant t \leqslant T; r(t) > \frac{f(t)}{\ln t}\big\} = o(T^\beta)$$ as $T \rightarrow \infty$, for some $0 \leqslant \beta < 1$ and some $f(t) = o(1)$. The stated condition is weaker than those in common use, and yet it is shown to imply the same limit theorems on the distribution of the maximum of the process. Examples are given of processes which satisfy the new condition and not the previous ones.

Let $\beta_T = (2a_T\lbrack\log(T/a_T) + \log\log T\rbrack)^{-\frac{1}{2}}, 0 < a_T \leqslant T < \infty$ and $\{W(t); 0 \leqslant t < \infty\}$ be a standard Wiener process. This paper studies the almost sure limiting behaviour of $\sup_{0\leqslant t\leqslant T-a_T} \beta_T|W(t + a_T) - W(t)|$ as $T \rightarrow \infty$ under varying conditions on $a_T = c \log T, c > 0$, the Erdos-Renyi law of large numbers for the Wiener process. A number of other results for the Wiener process also follow via choosing $a_T$ appropriately. Connections with strong invariance principles and the P. Levy modulus of continuity for $W(t)$ are also established.

Let $w_{jk}$ be a doubly indexed sequence of weights, let $\{X_k\}$ be a sequence of independent random variables and let $Q_n = \Sigma^n_{j,k=1} w_{jk}X_jX_k$. Sufficient conditions for the almost sure stability of $Q_n$ are given and the "tightness" of these conditions is investigated. These quadratic forms are weighted sums of dependent variables; however, their stability properties are very much like those established in the literature for weighted sums of independent variables.