Let $(X, \mathscr{A}, P)$ be a probability space and $\mathscr{C}$ a collection of measurable sets. Suppose $\mathscr{C}$ is a Donsker class, i.e., the central limit theorem for empirical measures holds uniformly on $\mathscr{C}$, in a suitable sense. Suppose also that suitable ($P\varepsilon$-Suslin) measurability conditions hold. Then we show that the $\log\log$ law for empirical measures, in the Strassen-Finkelstein form, holds uniformly on $\mathscr{C}$.

Let $X_1, X_2, \cdots$ be independent and identically distributed random variables, and let $M_n$ and $m_n$ denote respectively the mode and median of $\Sigma^n_1X_i$. Assuming that $E(X^2_1) < \infty$ we obtain a number of limit theorems which describe the behaviour of $M_n$ and $m_n$ as $n \rightarrow \infty$. When $E|X_1|^3 < \infty$ our results specialize to those of Haldane (1942), but under considerably more general conditions.

Necessary (and sometimes sufficient) conditions are given for a multi-step Markovian joining of two processes to attain the $\bar{d}$-distance between them. The conditions are applied to Markovian joinings of two-state Markov processes.

The equilibrium states and the time asymptotic behavior of Spitzer's zero-range interaction scheme are studied in the case of a positive-recurrent, irreducible single particle law and an attractive speed function. It is shown that within the natural phase space of finitely many particles per cell, clusters occur which lead time-asymptotically to infinite occupancy by certain individual cells. These cells are then identified in terms of the parameters of the model.

An urn contains $N$ objects, labelled with the integers $1, \cdots, N$. One object is removed at a time, without replacement. If after $n$ draws the largest number which has been observed is $m_n$, and the process is terminated, we receive a payoff $f(n, m_n)$. For payoff functions $f$ in a certain class, the optimal time to stop is with draw $$\tau_f = \inf\{n \geqslant 0: m_n - n \geqslant j_n\}$$ where the $j_n$ are computable from a simple algorithm, which permits also exact computation of the value $$V_f = E\{f(\tau_f, m_{\tau_f})\}.$$ We also study the behavior of $V_f$ when $N$ is large in special cases.

Using Laplace transforms, necessary and sufficient conditions are given for classes of life distributions which are of interest in reliability. Using these conditions, the problem of class closure under convolution is discussed. Furthermore, converses to results of A-Hameed for gamma wear processes are given.

Let $X_1, X_2, \cdots$ be identically distributed random variables defined on a probability space $(\Omega, \mathscr{F}, P)$ such that $E|X_1| < \infty$ and let $\mathscr{F}_0 \subset \mathscr{F}_1 \subset \cdots$ be nondecreasing sub-$\sigma$-algebras of $\mathscr{F}$ such that $X_n$ is $\mathscr{F}_n$-measurable for $n \geqslant 1$. Define $S_n = X_1 + \cdots + X_n$ and $\xi_n = E(X_n \mid \mathscr{F}_{n-1})$. The convergence and divergence of the series $\Sigma^\infty_{n=1} \operatorname{sgn}(S_{n+k})|S_{n+k}|^{-\alpha}(X_n - \xi_n)$, where $\alpha$ is a real number and $k$ a nonnegative integer, is considered and related to that of martingale transforms. This paper answers a question raised by Kai Lai Chung.

Let $\{X_k, k \geqslant 1\}$ be a stationary Gaussian sequence with $EX_1 = 0, EX^2_1 = 1$, and $EX_1X_{n+1} = r_n$. Let $c_n = (2 \ln n)^{\frac{1}{2}}, b_n = c_n - \ln(4\pi \ln n)/2c_n$ and set $M_n = \max_{1\leqslant k\leqslant n}X_k, \bar{X}_n = \frac{1}{n} \Sigma^n_{k=1}X_k$, and $s^2_n = \frac{1}{n} \Sigma^n_{k=1}(X_k - \bar{X}_n)^2$. If $r_n$ is not identically one and $(\ln n)/n\Sigma^n_{k=1}|r_k - r_n| = o(1)$, it is shown that \begin{equation*}\tag{1}\lim_{n\rightarrow\infty}P\big\{c_n\big(\frac{M_n - \bar{X}_n}{s_n} - b_n\big) \leqslant x\big\} = \exp\{-e^{-x}\}.\end{equation*} If we further assume $(r_n \ln n)^{-1} = o(1)$ then it is shown that \begin{equation*}\tag{2} \lim_{n\rightarrow\infty}P\big\{r^{-\frac{1}{2}}_n\big(\frac{M_n}{(1 - r_n)^{\frac{1}{2}}} - b_n\big) \leqslant x\big\} = (\frac{1 -\gamma}{2\pi})^{\frac{1}{2}}\int^x_{-\infty} e^{-\frac{(1 - \gamma)u^2}{2}}du\end{equation*} where $\gamma = F(\{o\})$ is the atom at zero of the spectral distribution associated with $r$. A version of these results for continuous time processes is also presented.

Mittal and Ylvisaker have shown a law of iterated logarithms for the maxima of certain stationary Gaussian processes. Their result is extended in this paper in that it is shown to be valid under weaker mixing conditions.

Let $\{X_{n,j}: j = 1, \cdots, n, n \geqslant 1\}$ be an array of nonnegative random variables in which each row forms a (finite) stationary sequence. The main theorem states sufficient conditions for the convergence of the distribution of the row sum $\Sigma_jX_{n,j}$ to a compound Poisson distribution for $n \rightarrow \infty$. This is applied to a stationary Gaussian process: it is shown that under certain general conditions the time spent by the sample function $X(s), 0 \leqslant s \leqslant t$, above the level $u$ may be represented as a row sum in a stationary array, and so has, for $t$ and $u \rightarrow \infty$, a limiting compound Poisson distribution. A second result is an extension to the case of a bivariate array. Sufficient conditions are given for the asymptotic independence of the component row sums. This is applied to the times spent by $X(s)$ above $u$ and below $-u$.

Excursion theory is applied to get identities for continuous time ladder variables. The identities are used to classify coharmonic and coinvariant functions for one dimensional Levy processes.

We attempt to relate the domains of partial attraction for sums of i.i.d. random variables to some properties of a generalised regular variation. We give a new characterisation of the domain of partial attraction of the normal distribution and new sufficient conditions for a distribution to belong to a domain of partial attraction.

Let $X_1, X_2, \cdots$ be a sequence of independent, identically distributed, random variables with a common distribution function $F. S_n$ denotes $X_1 + \cdots + X_n$. An increasing sequence of positive integers $(n_i)$ is defined to belong to $\mathscr{N}(F)$ if there exist normalizing sequences $(b_k)$ and $(a_k)$, with $a_k \rightarrow \infty$, so that every subsequence of $(a^{-1}_{n_i} S_{n_i} - b_{n_i})$ has a further subsequence converging in distribution to a nondegenerate limit. The main concern here is a description of $\mathscr{N}(F)$ in terms of $F$. This includes also conditions for $\mathscr{N}(F)$ to be void, as well as for $(1, 2, \cdots)\in \mathscr{N}(F)$, thus improving on some classical results of Doeblin. It is also shown that if there exists a unique type of laws so that $F$ is in the domain of partial attraction of a probability law if and only if the law belongs to that type, then in fact $F$ is in the domain of attraction of these laws.

For $d$-dimensional Brownian motion starting at 0 and having constant drift it is shown that the interarrival times between successive concentric spheres and the hitting place on the outermost sphere are independent. This generalizes results of Kent and Stern. A discrete analogue extending Samuels' theorem is indicated.

The purpose of this note is to show that the set of $L^1$-bounded "martingales in the limit", unlike the set of $L^1$-bounded "amarts", is not a vector lattice.

A local convergence theorem for a class of martingales with multidimensional time parameter is proved. This is accomplished by an elaboration of stopping time arguments used in the one-dimensional setting.

It is shown that for a random walk $\{S_n\}$ starting at the origin having generic step random variable $X$ with finite second moment and positive mean $\lambda^{-1} = EX$, the renewal function $U(y) = E {\tt\#}\{n = 0,1, \cdots: S_n \leqslant y\}$ satisfies for $y \geqslant 0$ $$|U(y) - \lambda y - \frac{1}{2}\lambda^2EX^2| \leqslant \frac{1}{2}\lambda^2EX^2 - \lambda EM \leqslant \frac{1}{2}\lambda^2EX^2_+$$ where $M = - \inf_{n\geqslant 0}S_n$. Both the upper and lower bounds are attained by simple random walk. Bounds are also given for $U(-y)(y \geqslant 0)$ and for the renewal function of a transient renewal process when $\Pr\{X \geqslant 0\} = 1 > \Pr\{0 \leqslant X < \infty\}$. The proof uses a Wiener-Hopf like identity relating $U$ to the renewal functions of the ascending and descending ladder processes to which Lorden's tight bound for the renewal process case is applied.

Let $\{\mu_1, \mu_2, \cdots\}$ be chosen from a strictly stationary, ergodic sequence of random variables each with distribution concentrated on $(0, \infty)$. Let $S_n = T_1 + \cdots + T_n$ be a sum of independent random variables where $T_j$ is exponential with mean $\mu_j$. Limiting properties of $S_n$ are considered. More limiting properties are derived under the assumption that $\{\mu_1, \mu_2, \cdots\}$ is strongly mixing and then under the assumption of independence.

Let $\{t_k\}$ be a sequence of points in $d$-dimensional Euclidean space. Let $\{X_k\}$ be a sequence of random variables with zero mean, i.i.d. or nearly so. If $\mathscr{A}$ is a class of subsets of $R^d$, let $$M_n(\omega) = \sup_{A\in\mathscr{A}}\Sigma_{\{k\leqslant n: t_k \in A\}}X_k(\omega).$$ $M_n$ is related to a commonly used estimator in monotone regression. Under various conditions on $\mathscr{A}$ and the points $\{t_k\}$, we study the a.s. convergence to zero of $M_n/n$ as $n \rightarrow \infty$.

The only strictly stationary random sequences satisfying a certain mixing condition are the sequences of independent, identically distributed rv's.