Consider the infinite particle system on the countable set $S$ with the simple exclusion interaction and one-particle motion determined by the stochastic transition matrix $p(x, y)$. In the past, the ergodic theory of this process has been treated successfully only when $p(x, y)$ is symmetric, in which case great simplifications occur. In this paper, coupling techniques are used to give a complete description of the set of invariant measures for the system in the following three cases: (a) $p(x, y)$ is translation invariant on the integers and has mean zero, (b) $p(x, y)$ corresponds to a birth and death chain on the nonnegative integers, and (c) $p(x, y)$ corresponds to the asymmetric simple random walk on the integers.

Let $\{X_n\}$ be a stationary Gaussian sequence with $EX_0 = 0, EX_0^2 = 1$ and $EX_0X_n = r(n)$. Let $c_n = (2 \ln n)^\frac{1}{2}$ and set $M_n = \max_{0\leqq k \leqq n} X_k$. It is presently known that if $r(n) \ln n = O(1)$, \begin{equation*}\tag{1}\lim \inf\frac{2c_n(M_n - c_n)}{\ln \ln n} = -1 \quad \text{and}\quad \lim \sup\frac{2c_n(M_n - c_n)}{\ln \ln n} = 1\end{equation*} with probability 1. Related results are obtained here assuming $r(n) = o(1)$ and $(r(n) \ln n)^{-1}$ is monotone for large $n$ and $o(1)$. Subject to some regularity in $r(n)$, it is shown that if $r(n) \ln n/(\ln \ln n)^2 = o(1)$, then a.s. \begin{equation*}\tag{2}\lim \inf\frac{2c_n(M_n - (1 - r(n))^{\frac{1}{2}}c_n - Z_n)}{\ln \ln n} = -1 \quad \text{and}\end{equation*} $$\lim \sup\frac{2c_n(M_n - (1 - r(n))^{\frac{1}{2}}c_n - Z_n)}{\ln \ln n} = 1$$ where $Z_n$ is the minimum variance estimate of the mean based on $X_0,\cdots, X_n$. Futhermore if $(\ln \ln n)^2/r(n) \ln n = o(1)$, then a.s. \begin{equation*}\tag{3}\lim_{n\rightarrow\infty} r^{-\frac{1}{2}}(n)(M_n - (1 - r(n))^{\frac{1}{2}}c_n - Z_n) = 0.\end{equation*} It is pointed out that (2) and (3) contain laws for $M_n$ which more closely resemble the one given here in (1). Corresponding results for continuous parameter Gaussian processes are sketched.

Let $\{X_n\}^{+\infty}_{-\infty}$ be a stationary sequence of random variables, with common distribution $\pi(dx)$. If the initial value $X_0$ is repeated with probability one (e.g. when $\pi(dx)$ is discrete), then the "shifted" sequence $\{X_{n+N}\}^\infty_{-\infty}$ is also stationary where $N = N(\omega)$ is the first $n > 0$ for which $X_n(\omega) = X_0(\omega)$. Surprisingly, this may even occur when $\pi(dx)$ is continuous and $\{X_n\}$ is ergodic (although not when $\{X_n\}$ is $\phi$-mixing). For Markov sequences, we also give other conditions which prohibit the a.s. recurrence of $X_0$. For recurrent sequences, we show that when $X_0$ is "conditionally discrete," the invariant $\sigma$-field for the $\{X_{n+N}\}$ process coincides (up to null sets) with $X_0 \vee \mathscr{A}$, the $\sigma$-field generated by $X_0$ and the invariant sets for $\{X_n\}$. Finally, we find an expression for $E(N\mid X_0 \vee \mathscr{A})$ which reduces to Kac's recurrence formula when $X_0$ is an indicator function.

We consider an age-dependent model which not only allows the generating function to be age-dependent, but also allows the parent to reproduce several times during its lifetime. By using the notion of $\vee$-space-time harmonic functions, we study the behavior of $Z_t$, the number of particles alive at time $t$, in the supercritical case. In particular we obtain results which are analogous to the classically known results for the Bellman-Harris model; in fact, we obtain convergence in probability.

We show that there always exists a sequence of normalizing constants for the supercritical multitype Galton-Watson process so that the normalized sequence converges in probability to a limit which is proper and not identically zero. The Laplace-Stieltjes transform of the limit random variable is characterized as the unique solution under certain conditions of a vector Poincare functional equation.

Semimarkov processes with discrete state spaces are considered without restrictions on their probability laws. They admit states where every visit lasts a positive time even though there may be infinitely many such visits in a finite interval. These are called unstable holding states as opposed to the stable holding states which are encountered in Markov processes. Further, it is possible to have instantaneous states at which the behavior is that at an ordinary instantaneous state of a Chung process, or that at a sticky boundary point, or that at a nonsticky point. To convert such processes to Chung processes, each unstable state is split into infinitely many stable ones, and then a random time change is effected whereby some sets of constancy are dilated, and sojourn intervals are altered to have exponentially distributed lengths.

An infinite sequence of rankable individuals (rank $1 =$ best) arrive at times which are i.i.d., uniform on (0, 1). We, in effect, observe only their relative ranks as they arrive. We seek a stopping rule to minimize the mean of a prescribed positive increasing function, $q(\bullet)$, of the actual rank of the individual chosen. Let $f(t)$ be the minimal mean among all stopping rules which are greater than $t$. Then $f(\bullet)$ is a solution to a certain differential equation which is derived and used to find an optimal stopping rule. This problem is in a strong sense the "limit" of a corresponding sequence of "finite secretary problems" which have been examined by various authors. The limit of the "finite-problem" minimal risks is finite if and only if the differential equation has a solution, $f(\bullet)$, which is finite on $\lbrack 0, 1)$ with $f(1^-) = \sup q(n)$. Usually, if such a solution exists, it is unique, in which case $f(0)$ is both the minimal risk for the infinite problem and the limit of the "finite-problem" minimal risks.

A new technique is developed for studying the convergence of nets of probability measures on a topological group. It is applied to results concerned with the interplay between convergence and convolutions of measures like properties of the convolution mapping, divisibility of measures and convolution semigroups. Our method gives a unified and simple approach to these results.

A lemma concerning real sequences is proved and applied to sequences of random variables $(\mathrm{rv}) X_1, X_2\cdots$ to determine conditions under which $\lim\sup_{n\rightarrow\infty} b_n^{-1} \sum^n_{m=1} f(m/n)X_m < \infty$ a.s. for all $f$ in a particular collection of absolutely continuous functions and for nondecreasing positive real sequences $\{b_n\}$. Theorems in the case $b_n = (2n \log \log n)^\frac{1}{2}$ are proved for generalized Gaussian rv, for equinormed multiplicative systems and for certain martingale difference sequences.

We prove that a Gaussian Taylor series has the recurrence property on any rectifiable Jordan domain such that the intersection of its boundary with the unit circle is of positive measure.

A sequence $\{Y_n\}^\infty_{n=1}$ of random variables with values in a metric space is mixing with limiting distribution $G$ if $P(\{Y_n \in A\}\mid B) \rightarrow G(A)$ for all $G$-continuity sets $A$ and all events $B$ that have positive probability. It is shown that if $\{Y_n\}$ is mixing with limiting distribution $G$ and if the support of $G$ is separable, then the range $\{Y_n(\omega); n \geqq 1\}$ is dense in the support of $G$ almost surely. A theorem that, under rather general conditions, establishes mixing for the summation processes based on a martingale is given, and as an application it is shown that, under certain conditions, the range of the periodogram is dense in $R^+$ almost surely.

We discuss analogues of Doob's convergence theorem for submartingales with values in Banach lattices with the Radon-Nikodym property.

Lamperti in 1964 showed that the convergence of the marginals of an extremal process generated by independent and identically distributed random variables implies the full weak convergence in the Skorohod $J_1$-topology. This result is generalized to the $k$th extremal process and to random variables which need not be identically distributed. The proof here is based on the weak convergence of a certain point-process (which counts the number of up-crossings of the variables) to a two-dimensional nonhomogeneous Poisson process.

Weak convergence of a sequence of two-dimensional time parameter stochastic processes constructed from partial sums of induced order statistics to a standard Brownian sheet process is established.

Let $\{X_k: k \geqq 1\}$ be a sequence of i.i.d.rv with $E(X_i) = 0$ and $E(X_i^2) = \sigma^2, 0 < \sigma^2 < \infty$. Set $S_n = X_1 + \cdots + X_n$. Let $Y_n(t)$ be $S_k/\sigma n^\frac{1}{2}$ for $t = k/n$ and suitably interpolated elsewhere. This paper gives a generalization of a theorem of Iglehart which states weak convergence of $Y_n(t)$, conditioned to stay positive, to a suitable limiting process.

Let $\{X_n\}$ be a sequence of independent and identically distributed random variables with $EX_1 = 0$. Suppose that there exists a constant $a > 0$, such that $Z_n = (an^r)^{-1}(X_1 + X_2 + \cdots + X_n)$ converges in law to a stable distribution function (df) $V(x)$ as $n \rightarrow \infty$. If, in addition, we assume that the characteristic function of $X_1$ is absolutely integrable in $m$th power for some integer $m \geqq 1$, then for all large $n$, the df $F_n$ of $Z_n$ is absolutely continuous with a probability density function (pdf) $f_n$ such that the relation $$\lim_{n\rightarrow\infty}|x\|f_n(x) - \nu(x)| = 0$$ holds uniformly in $x, -\infty < x < \infty$, where $v$ is the pdf of $V$.

The usual form of local limit theorem is extended to an arbitrary supercritical Galton-Watson process with arbitrary initial distribution. The existence of a continuous density on $(0, \infty)$ for the limit random variable $W$, in the process initiated by a single ancestor, follows from the derivation.

The potential of a natural additive functional of a transient standard process is represented as a potential of a measure without the usual assumption of strong duality for the process. The balavage on a Borel set, $B$, of the potential of an additive functional or bounded function is represented as the potential of a measure supported by the closure of $B$.

Let $f$ be a measurable convex function from $R^k$ to $R^1$ and let $X_1, \cdots, X_k$ be real-valued integrable random variables. The best approximation for $f(EX_1, \cdots, EX_k)$ one can get by Jensen's inequality is $f(EX_1, \cdots, EX_k) \leqq \inf Ef(\mathbf{Z})$ where the infimum is taken over all $k-\dim$. random vectors $\mathbf{Z} = (Z_1, \cdots, Z_k)'$ such that $Z_i$ has the same distribution as $X_i (1 \leqq i \leqq k)$. An application is given in the case where $f(y)$ is the span of the vector $y$ which leads to a new approximation for $f(A\mathbf{u})$ where $A$ is a stochastic $(k \times m)$-matrix and $\mathbf{u}$ is an arbitrary element of $R^m$.

A real number $t$ is an admissible translate of a probability $\varphi$ if $\varphi (A) = 0$ implies that $\varphi_t(A) \equiv \varphi (A - t) = 0$. Conditions are given on its set of admissible translates which ensure that $\varphi$ has a density. The theorems also describe the set where the density is positive and contain as a corollary the result that if $\varphi$ is not absolutely continuous, then the set of admissible translates has an empty interior.