It is now ten years since Hammersley and Welsh discovered (or invented) subadditive stochastic processes. Since then the theory has developed and deepened, new fields of application have been explored, and further challenging problems have arisen. This paper is a progress report on the last decade.

Here is another way to prove Levy's conditional form of the Borel-Cantelli lemmas, and his strong law. Consider a sequence of dependent variables, each bounded between 0 and 1. Then the sum $S$ of the variables tends to be close to the sum $T$ of the conditional expectations. Indeed, the chance that $S$ is above one level and $T$ is below another is exponentially small. So is the chance that $S$ is below one level and $T$ is above another. The inequalities also show that for a sequence of dependent events, such that each has uniformly small conditional probability given the past, and the sum of the conditional probabilities is nearly constant at $a$, the number of events which occur is nearly Poisson with parameter $a$.

In this paper criticality-limit theorems are proved for a two-type branching process in which cancellation of particles of opposite type occurs after reproduction.

A multitype branching process with immigration is considered. A necessary and sufficient condition is found for when the process has a limiting stationary distribution. As a corollary to this result we are able to give necessary and sufficient conditions for when the extinction time for an ordinary multitype process has finite expectation.

Strassen's law of the iterated logarithm for Brownian motion is extended to a class of Gaussian processes. Let $\{X(t), t \geqq 0\}$ be a real continuous Gaussian process with $X(0) = 0$, mean zero and continuous covariance kernel $R(s, t)$. Define a random sequence $\{f_n(t, \omega)\}$ in $C\lbrack 0, 1 \rbrack$ by $f_n(t, \omega) = X(nt, \omega)/(2R(n, n) \log \log n)^{\frac{1}{2}}$. Under certain conditions on $R$ it is shown that with probability one $\{f_n(t, \omega)\}$ is equicontinuous and the set of its limit points is the unit ball of the reproducing kernel Hilbert space with reproducing kernel $\Gamma$ determined by $R$. The result generalizes the author's earlier result (1972).

This paper is mainly a survey of results on the problem of finding necessary and sufficient conditions for a Gaussian process to be continuous. The relationship between this problem and the same one for random Fourier series is explored. Some new results are presented that give continuity conditions for stationary Gaussian processes in terms of the spectrum of the process. Let $X(t)$ be a real-valued stationary Gaussian process; $EX(t) = 0, EX^2(t) = 1$. Define $F$ by the equation $EX(t + h)X(t) = \int^\infty \cos \lambda h dF(\lambda)$. Assume that $F(\lambda)$ is concave for $\lambda \geqq\lambda_0 > 0$ then $X(t)$ is continuous a.s. if and only if $$\int^\infty \frac{(1 - F(x))^{\frac{1}{2}}}{x(\log x)^{\frac{1}{2}}} dx < \infty.$$ A similar result holds for Fourier series with normal coefficients.

Suppose $\mu$ is a positive measure on the half-line $\lbrack 0, \infty)$, of total mass $m, \Phi$ is the sum of a power series with nonnegative coefficients which converges at the point $m$, and $\lambda$ is the measure on $\lbrack 0, \infty)$ whose Fourier transform $\hat{\lambda}$ is $\Phi(\hat{\mu})$. The lower limit of the ratios $\lambda(\lbrack s, \infty))/\mu(\lbrack s, \infty))$, as $s\rightarrow\infty$, is compared to the number $\Phi'(m)$, under a variety of conditions.

Ergodic behavior for Markov chains can be determined by studying the properties of the corresponding sequence of stochastic transition kernels. Dobrushin's ergodic coefficient has been useful for this purpose. In this paper we define pointwise strongly and weakly ergodic behavior for sequences of nonnegative kernels and use Dobrushin's ergodic coefficient to give sufficient conditions for these two types of behavior. Applications are given to sequential probability ratio tests.

Consider a reversible (self-adjoint) Markov process with a discrete time parameter and stationary transition probability functions satisfying the Harris recurrence condition. $\mathbf{P}^{(n)}(x, S)$ denotes the $n$-step transition probability function from $x$ to the measurable set $S$ and $\pi$ is the sigma-finite stationary measure induced by the above hypotheses. Using both a functional analytic representation for reversible probabilities and probabilistic identities, various limits are considered for both general and discrete spaces. The principle result gives necessary and sufficient conditions for sets $A$ and $\mathbf{B}$ so that a reversible, aperiodic Markov process satisfies the strong ratio limit property $\lim_{m\rightarrow\infty} \mathbf{P}^{(n + k)} (\mu, \mathbf{A})/\mathbf{P}^{(n)} (\nu, \mathbf{B}) = \pi(\mathbf{A})/\pi(\mathbf{B}$ where $\mu$ and v are arbitrary probability distributions defined on the space and $k$ is any integer.

Let $X(t), X(0) = 0$, be a nonsingular diffusion in the natural scale in a neighborhood of 0, and let $\mathbf{f}(t, x)$ be its local time. The local behavior of $X(\mathbf{f}^{-1}(t, 0))$ is studied, and used to obtain upper and lower functions of a new type for $X(t)$ at $t = 0$.

The precise asymptotic behavior of certain expected first passage times plays an important role in C. Stone's theory of weak convergence of Markov processes. For a special class of random walks studied by Harris (1952), Lamperti (1962) and Karlin-McGregor (1959) we present a new method, using a maximum principle for a linear second order difference operator, that yields these asymptotic estimates. As a corollary we obtain an alternative proof of Lamperti's (1962) invariance principle.

The main results are functional central limit theorems for superpositions of randomly selected partial sums in which the random variables being summed are independent and have distributions in the domain of attraction of stable laws. These results extend those of Tucker and Sreehari concerning when convolutions of distributions are attracted to stable laws. Other functional central limit theorems are presented for more general sums. The results herein extend the central limit theory for additive processes on Markov chains.

A simple proof, with a little-known extension, of a density version of Karamata's Tauberian theorem is presented, and the result applied to limit distributions of the Galton-Watson process.

In this note we show that the Littlewood-Paley argument also applies to the martingale square function.

In this note a weak convergence result in the Skorohod space $D^2\lbrack a, b \rbrack$ for a sequence of stochastic processes generated by the sample extrema of a stationary Gaussian sequence is obtained.

For the supercritical branching process with random environments, the rate of growth of the generation size $Z_n$ is studied in the marginal distribution. It is shown that unless the environmental process yields a constant conditional expectation $E(Z_1 \mid \zeta)$, the asymptotic distribution of $$(Z_n \exp(-nE_\zeta(\log E(Z_1 \mid \zeta))))^{n^{-\frac{1}{2}}}$$ is that of $Ue^V$ where $U$ and $V$ are independent, $P(U = 0) = 1 - P(U = 1) = P(Z_n \rightarrow 0)$ and $V$ is normal $(0, V_\zeta(\log E(Z_1\mid \zeta))$.

Let $\{X_i\}^\infty_{i = -\infty}$ be a sequence of independent random variables with zero means and let $P\lbrack |X_i| \geqq x \rbrack \leqq M \int^\infty_x \exp\{-\Upsilon t^2\} dt$ for all $x \geqq 0$ where $M$ and $\Upsilon$ are positive constants. Let $((a_{ij}))^\infty_{i, j = -\infty}$ be an infinite matrix of real numbers with $a_{ij} = a_{ji}$ for all $i, j$ and $\Lambda^2 = \sum_{i,j}a^2_{ij} < \infty$. Let $\|A\|$ be the norm of $A = ((|a_{ij}|))$ considered as an operator on $l_2$ and set $S = \sum_{i,j}a_{ij}(X_iX_j - E(X_i X_j))$. In this note it is shown that there exist positive constants $C_1$ and $C_2$ depending only on $M$ and $\Upsilon$ such that $P\lbrack S \geqq \varepsilon \rbrack \leqq \exp \{-\min (C_{1 \varepsilon}/\|A\|, C_{2\varepsilon^2}/\Lambda^2)\}$ for all $\varepsilon > 0$. This result has previously been established in the literature for sequences of random variables which have symmetric distributions.

An estimate of the expected value of the sum of independent identically distributed random variables in the domain of normal attraction of a stable law of index $\alpha$ is obtained. This estimate is then used to obtain a generalization of the Helly-Bray lemma.