Let $X(t)$ be a real stationary Gaussian process with covariance function $r(t);$ and let $f(t), t \geqq 0,$ be a nonnegative continuous function which vanishes only at $t = 0.$ Under certain conditions on $r(t)$ and $f(t),$ we find, for fixed $T > 0$ and for $u \rightarrow \infty$ (i) the asymptotic form of the probability that $X(t)$ exceeds $u + f(t)$ for some $t \in \lbrack 0, T \rbrack;$ and (ii) the conditional limiting distribution of the time spent by $X(t)$ above $u + f(t), 0 \leqq t \leqq T,$ given that the time is positive.

Let $\{y(\tau), 0 \leqq \tau \leqq 1\}$ be a sample continuous Gaussian process, and let $T\lbrack y, \alpha \rbrack$ denote the time that $y(\cdot)$ spends above level $\alpha:$ $$T\lbrack y, \alpha \rbrack = \int^1_0 V(y(\tau) - \alpha) d\tau,$$ where $V(x) = 0$ or 1 according as $x \leqq 0$ or $x > 0.$ In this paper it is proved that, as $\alpha \rightarrow \infty,$ $$P\{T\lbrack y, \alpha \rbrack > \beta\} = \exp \{-(\alpha^2/2)\mathbf{k}_\beta(1 + o(1))\}$$ where $k_\beta$ is a particular functional of the covariance function of the process.

Let $\{X_n, n = 0, \pm 1, \pm 2, \cdots\}$ be a stationary Gaussian stochastic process with means zero, variances one, and covariance sequence $\{r_n\}$. Let $M_n = \max \{X_1, \cdots, X_n\}$ and $S_n =$ second largest $\{X_1, \cdots, X_n\}$. Limit properties are obtained for the joint law of $M_n$ and $S_n$ as $n$ approaches infinity. A joint limit law which is a function of a double exponential law is known to hold if the random variables $X_i$ are mutually independent. When $M_n$ alone is considered Berman has shown that a double exponential law holds in the case of dependence provided either $r_n \log n \rightarrow 0$ or $\sum^\infty_{n=1} r_n^2 < \infty$. In the present work it is shown that the above conditions are also sufficient for the convergence of the joint law of $M_n$ and $S_n$. Weak convergence properties of the stochastic processes $M_{\lbrack nt\rbrack}$ and $S_{\lbrack nt \rbrack}$ with $0 < a \leqq t < \infty$ are also discussed.

An account of known upper and lower bounds for standard diagonal Markov transition functions and standard $p$-functions taking on a given value at time $t$ is presented, and some new results are derived.

Asymptotic forms for the optimal expected payoffs (minimal costs) for a generalized class of "Secretary" problems are investigated by an analysis of a related family of differential equations. A class of unbounded payoff functions with bounded expected payoffs is determined and methods for generating the expected payoffs are developed.

The main result in this paper is an invariance principle for the law of the iterated logarithm for square integrable martingales subject to fairly mild regularity conditions on the increments. When specialized to the case of identically distributed increments the result contains that of Stout [16] as well as the invariance principle for independent random variables of Strassen [17]. The martingale result is also used to obtain an invariance principle for the iterated logarithm law for a wide class of stationary ergodic sequences and a corollary is given which extends recent results of Oodaira and Yoshihara [10] on $\phi$-mixing processes.

For certain recurrent diffusion processes, Motoo has given an integral test which allows one to determine whether an increasing function belongs to the upper or lower class relative to the process at hand. We show that a refinement of his methods yields asymptotic estimates for the tail probabilities of the time of last crossing of an upper class function $g$ in cases where the speed measure of the process has sufficiently thin tails and the curve $g$ increases sufficiently slowly. Similar results are derived for certain extremal processes and for non-decreasing stable processes.

A branching process is considered for which the conditional distributions of the litter sizes, given the past, are allowed to vary from period to period and are required only to belong to some set $\mathscr{M}$. The process is non-Markovian in general. For various interesting $\mathscr{M}$, bounds are derived on (i) the probability of extinction, (ii) the mean time to extinction, (iii) the probability that a generation size exceeds a given number, (iv) the expected maximum generation size, and (v) the mean total population size. In (i), (ii) and (v), the optimal strategies which achieve the bounds are identified. The techniques used are similar to those used in the theory of gambling as developed by Dubins and Savage (How to Gamble if You Must, McGraw-Hill (1965)).

Some results of the paper "Inequalities for Branching Processes" [Ann. Prob. 1 (1973)] by the same author are extended to a multitype branching process. Bounds are obtained on the probability of extinction and mean time to extinction of the process when the probability transition laws are allowed to vary from period to period and are required only to belong to some class $\mathscr{M}$.

Certain mixing sequences of dependent `rare' events are considered and a Poisson limit is established for the probability that $k$ events occur. An asymptotic distribution for the number of upcrossings of a high level by certain stochastic processes is considered as an application.

Fine gives axioms on a binary relation $\precsim$ on a field of events, with $A \precsim B$ interpreted as "$A$ is (subjectively) no more probable than $B$," sufficient to guarantee the existence of an order-preserving probability measure and an additive order-preserving probability measure. It is noted that one of Fine's axioms, that the order topology have a countable base, can be replaced by the more appealing axiom that there is a countable order-dense subset.

An almost sure invariance principle for Kolmogorov-Smirnov statistics for vector chance variables is established along the lines of Theorems 1.4 and 4.9 of Strassen [Proc. Fifth Berkeley Symp. Math. Statist. Prob. (1967) 2 315-343]. This strengthens certain asymptotic expressions on the probability of moderate deviations for Kolmogorov-Smirnov statistics, obtained earlier by Gnedenko, Karoluk and Skorokhod, and by Kiefer and Wolfowitz, among others.

We show that the $L_1$ norm of the difference between the standard normal distribution and the distribution of the standardized sum of $n$ independent random variables is less than 72 $R_n$, where $R_n$ is a sum of standardized "inside" third and "outside" second moments. We conjecture that 72 can be replaced by 36 or even less. We also prove a similar result for $m$-dependent random variables, but no constant is specified.

Let $S_n$ denote the partial sums of i.i.d. random variables with mean zero and moment generating function existing in some neighborhood of the origin. We give explicit upper bounds for $P_m^+ = P(S_n \geqq a + bn$ for some $n \geqq m)$ and $P_m = P(|S_n| \geqq a + bn$ for some $n \geqq m), a \geqq 0, b > 0$. These bounds immediately give the rate of convergence for the strong law of large numbers. An application is also made to a sequential selection procedure.

The large deviation estimate, used in classical proofs of the law of the iterated logarithm for i.i.d. random variables, implies the random variables satisfy a condition more stringent than a finite variance. Thus it is impossible to prove the law of the iterated logarithm in its full strength (i.e. assuming only a finite second moment) by using such a deviation estimate in a "straightforward" manner.

This paper is concerned with the stochastic processes in Banach space arising from the solutions of stochastic integral equations. It is shown that under certain assumptions the corresponding measures of two such stochastic processes are equivalent. The Radon-Nikodym derivative is also given.

In this paper it is shown that the classical local conditions for sample continuity of random processes without discontinuities of the second kind may be replaced by weaker ones of a simpler form. When such a process is obtained as the limit in distribution in the Skorohod topology, the new conditions may be conveniently restated in terms of the sequence.