For an $n$-dimensional random field $X(\mathbf{t})$ we define the excursion set $A$ of $X(\mathbf{t})$ by $A = \{\mathbf{t} \in \mathbf{I}_0: X(\mathbf{t}) \geqq u\}$, where $I_0$ is the unit cube in $R^n.$ It is shown that the natural generalisation of the number of upcrossings of a one-dimensional stochastic process to random fields is via the characteristic of the set $A$ introduced by Hadwiger (1959). This characteristic is related to the number of connected components of $A$. A formula is obtained for the mean value of this characteristic when $n = 2, 3$. This mean value is calculated explicitly when $X(\mathbf{t})$ is a homogeneous Gaussian field satisfying certain regularity conditions.

Some two-dimensional time-parameter stochastic processes are constructed from a sequence of linear rank statistics by considering their projections on the spaces generated by the (double) sequence of anti-ranks. Under appropriate regularity conditions, it is shown that these processes weakly converge to Brownian sheets in the Skorokhod $J_1$-topology on the $D^2\lbrack 0, 1 \rbrack$ space. This unifies and extends earlier one-dimensional invariance principles for linear rank statistics to the two-dimensional case. The case of contiguous alternatives is treated briefly.

This paper examines the size, $Z(t)$, of a population as a function of time. $Z(t)$ is just like the ordinary Bellman-Harris age dependent branching process except that the number of daughters born to an individual in the $n$th generation is allowed to depend on $n$. The renewal theory of William Feller and Laplace transform theory are used to obtain the behavior of $EZ(t)$ as $t$ approaches infinity, and the convergence of $Z(t)/E(Z(t))$ in quadratic mean.

The age distribution for a supercritical Bellman-Harris process is proven to converge in probability to a deterministic distribution under assumptions slightly more than finite first moment. If the usual "$j \log j$" condition holds, then the convergence can be strengthened to hold w.p. 1. As a corollary to this result, the population size, properly normalized is shown to converge w.p. 1 to a nondegenerate random variable under the "$j \log j$" assumption.

Let $X_1, X_2, \cdots$ be i.i.d. random variables with $EX_1 = 0, EX_1^2 = 1$ and let $S_n = X_1 + \cdots + X_n$. In this paper, we study the ladder variable $S_N$ where $N = \inf \{n \geqq 1: S_n > 0\}$. The well-known result of Spitzer concerning $ES_N$ is extended to the higher moments $ES_N^k$. In this connection, we develop an asymptotic expansion of the one-sided moments $E\lbrack(n^{-\frac{1}{2}}S_n)^-\rbrack^\nu$ related to the central limit theorem. Using a truncation argument involving this asymptotic expansion, we obtain the absolute convergence of Spitzer's series of order $k - 2$ under the condition $E|X_1|^k < \infty$, extending earlier results of Rosen, Baum and Katz in connection with $ES_N$. Some applications of these results to renewal theory are also given.

Let $X_1, X_2, \cdots$ be i.i.d. with a finite positive mean $\mu$ and a finite positive variance $\sigma^2$ and let $S_n = X_1 + \cdots + X_n, n \geqq 1$. Further, let $0 \leqq \alpha < 1$ and $t_c$ be the first $n \geqq 1$ for which $S_n > cn^\alpha$ and let $W_c(a) = \sum^\infty_{n = 1} P\{t_c > n, c(n + 1)^\alpha - S_n < a\}$. Under some additional conditions on the distribution of $X_1$ we show that $W_c$ converges weakly to a limit $W$, where $W'(a) = \beta\mu^{-1}P\{S_k \geqq (k + 1)\alpha\mu - a$, for all $k \geqq 0\}$, with $\beta = 1/(1 - \alpha)$. We then find the asymptotic distribution of the excess $R_c = S_{t_c} - ct_c^\alpha$ and show that $R_c$ is asymptotically independent of $t_c^\ast = (t_c - E(t_c))/E(t_c)^{\frac{1}{2}}$, and we compute $E(t_c)$ up to terms which are $o(1)$ as $c \rightarrow \infty$.

For a general class of statistics expressible as extrema of certain sample functions, an almost sure invariance principle, particularly useful in the context of the law of iterated logarithm and the probabilities of moderate deviations, is established, and its applications are stressed.

A simple proof of a result that a point process subject to independent thinning converges to a Poisson process is given.

A variant of the ergodic maximal function is studied. This maximal function reduces to the usual one for $f \geqq 0$, but the cancellation between the positive and negative parts of $f$ causes interesting behavior. In particular the maximal function can be in $L^1$ even when the function is not in $L \log^+ L$. A relation between this maximal function and the ergodic Hilbert transform is studied.

Linear prediction theory for a stationary sequence $X_n$ ordinarily begins with the assumption that the covariance $R(n) = E(X_{m + n} \bar{X}_m) = \int^\pi_{-\pi} e^{i\lambda n} dF(\lambda)$ is known. The best linear predictor of $X_0$ given the past $X_{-1}, X_{-2}, \cdots$ is then the projection $\psi$ of $X_0$ on the span of $X_{-1}, X_{-2}, \cdots$. The prediction error is $E(|X_0 - \psi|^2)$. In practice $R$ is not known but is estimated from the past. If the process is ergodic and the entire past is known this causes no problem since then the estimate $\hat{R}$ of $R$ must equal $R$. But if the process is not ergodic then $\hat{R}$ does not equal $R$. In this paper we consider the relationship between prediction using $\hat{R}$ and $R$. One conclusion is that if the process is Gaussian, it doesn't matter whether $\hat{R}$ or $R$ is used in constructing the best linear predictor. The predictor is the same and the prediction error is the same.

For the distribution of the standardized sum of independent and identically distributed random variables, nonuniform central limit bounds are proved under an appropriate moment condition. From these theorems a condition on the sequence $t_n, n \in \mathbb{N}$, is derived which implies that $1 - F_n(t_n)$ is equivalent to the corresponding deviation of a normally distributed random variable. Furthermore, a necessary and sufficient condition is given for $1 - F_n(t_n) = o(n^{-c/2}t_n^{2 + c})$.

A sequence of integrable random variables $\{X_n, n \geqq 0\}$ is a weak martingale if $E(X_n \mid X_m) = X_m$ a.s. for all $0 \leqq m < n$. We present an example of a weak martingale which is not a martingale. It is bounded and has countably many paths.

The general renewal equation and real variable methods are used to show that for a renewal process with generic lifetime random variable $X \geqq 0$ having distribution $F$ and finite first and second moments $EX = \lambda^{-1}$ and $EX^2$, the renewal function $U(x) = \sum^\infty_0 F^{n^\ast(x)$ satisfies $U(x) \leqq \lambda x_+ + C\lambda^2EX^2$ for a certain constant $C$ independent of $F$. Stone (1972) showed that $1 \leqq C \leqq 2.847 \cdots$; it is proved here that $C \leqq 1.3186 \cdots$ and conjectured that $C = 1$.

Let $\{S_n: n \geqq 0\}$ denote the recurrent random walk formed by the partial sums of i.i.d. integer-valued random variables with zero mean and finite variance. Let $T = \min \{n \geqq 1: S_n = 0\}$. Our main result is an invariance principle for the random walk conditioned by the event $\lbrack T = n\rbrack$. The limiting process is identified as a Brownian excursion on [0, 1].

In 1972, R. E. Welsch obtained a class of two-dimensional laws which includes the class of all possible limit laws for a sequence of normalized pairs $(M_n, S_n)$, where $M_n$ and $S_n$ are the maximum and the second maximum of the first $n$ terms from a stationary strong-mixing sequence of random variables. In this note an example is given to show that these two classes are identical.

As an extension of the work of Denzel, Kemeny, and Snell on the excessive functions of a continuous time Markov chain, this paper introduces the concept of reconstructability in two forms. First, there is reconstructability from the class of excessive functions, where it is seen that the transition matrix for a transient chain with a finite atomic exit boundary can be written down knowing only the membership of its class of excessive functions. A similar result is true, with the transient condition dropped, for reconstructability from the characteristic operator, based on a natural extension to the boundary of the operator corresponding to the initial derivative matrix.

Let $X_s$ be a continuous martingale and $Q\nu$ be an increasing sequence of partitions of [0, 1]. Let $$S^2(Q_\nu) = \sum_{t_i\in Q_\nu} (X_{t_i} - X_{t_{i - 1}})^2.$$ An example is given in which $$\lim \sup_{\nu \rightarrow \infty} S^2(Q_\nu) = \infty.$$

Almost sure estimates of the rate of convergence for the supercritical Galton-Watson process are obtained, e.g. $W - W_n = o(m^{-n/q})$ a.s. if and only if $E(Z_1^p \mid Z_0 = 1) < \infty$, where $1 < p < 2, 1/p + 1/q = 1$. Extensions to the multitype and continuous time cases are outlined.

Let $\mathbf{P}_n \mid \mathscr{A}, n \in \mathbb{N}$, be a sequence of probability measures converging in total variation to the probability measure $\mathbf{P} \mid \mathscr{A}$ and $\mathscr{C}_n \subset \mathscr{A}, n \in \mathbb{N}$, be a sequence of $\sigma$-lattices converging increasing or decreasing to the $\sigma$-lattice $\mathscr{C}$. Then for every uniformly bounded sequence $f_n, n \in \mathbb{N}$, converging to $f$ in $\mathbf{P}$-measure we show in this paper that the conditional $p$-mean $\mathbf{P}_n^\mathscr{C}k f_j$ converge to $\mathbf{P}^\mathscr{C}f$ in $\mathbf{P}$-measure if $n, k, j$ tends to infinity. The methods used in this paper are completely different from those used to prove the corresponding result for $\sigma$-fields instead of $\sigma$-lattices.

Asymptotic expressions for the downcrossing probabilities of certain extremal and their related order statistics processes are obtained.