The asymptotic behaviour of the residual life time at time $t$ is investigated (for $t \rightarrow \infty$). We derive weak limit laws and their domains of attraction and treat rates of convergence and moment convergence. The presentation exploits the close similarity with extreme value theory.

A characterization of the capacity region of a two-way channel is given for the communication situation, in which both senders send independent messages simultaneously to both receivers and all senders and receivers are at different terminals.

We consider a linear time invariant model relating one process with stationary increments to another such process. The model contains the stationary $G/G/\infty$ queue and a bivariate cluster process as particular cases. The parameters of the model are shown to be identifiable through cross-spectral analysis and estimates are shown to be asymptotically normal under regularity conditions. In the case of the $G/G/\infty$ queue, the parameters considered are the characteristic function and the distribution function of the service time. The estimates are based on a stretch of entry and exit times for the system.

Particles perform independent random walks on the integers, and are annihilated if they cross paths or land at the same point. The problem is to determine whether the origin is hit infinitely often. The answer is shown to depend on the initial distribution of particles in accordance with a "log log law." Several equivalent models are mentioned.

This paper develops, for a Markov chain $\{X_n\}$ on a general space $(\mathscr{X}, \mathscr{F})$ with $n$-step transition probabilities $P^n(x, A), x \in \mathscr{X}, A \in \mathscr{F}$, a theory analogous to that of Vere-Jones for Markov chains on the integers. If the chain is $\phi$-irreducible there is a partition $\mathscr{K}$ of $\mathscr{X}$ such that $\phi$-almost all of the power series $G_z(x, A) = \sum_n P^n(x, A)z^n$ have a common radius of convergence $R$ for $A$ in any element of $\mathscr{K}$, and they all diverge ($R$-recurrence) or all converge ($R$-transience) for $z = R$. The $R$-recurrent case is then investigated, and it is shown that there exist essentially unique non-zero solutions $Q, f$ to the $R$-subinvariant equations $Q \geqq RQP$ and $f \geqq RPf$, and that $Q$ and $f$ satisfy these inequalities with equality: a relationship between $Q$ and $f$ and first-entrance probabilities is also established. Further, if $\{X_n\}$ is aperiodic, $\lim_{n\rightarrow\infty} R^nP^n(x, A) = f(x)Q(A)/\int_\mathscr{X} f(y)Q(dy)$ for almost all $x \in \mathscr{X}$ and $A$ in any element of a second partition. The methods used are probabilistic and depend mainly on generating function techniques: it is pointed out that these techniques do not depend on the substochasticity of the transition probabilities, and hence the results are true in a much wider context.

This paper is a sequel to a previous paper of similar title. The structure of $r$-subinvariant measures for a Markov chain $\{X_n\}$ on a general state space $(\mathscr{X}, \mathscr{F})$ is investigated in the $r$-transient case, and a Martin boundary representation is found. Under certain continuity assumptions on the transition law of $\{X_n\}$ the elements of the Martin boundary are identified when $\mathscr{F}$ is countably generated, and a necessary and sufficient condition for an $r$-invariant measure for $\{X_n\}$ to exist is found. This generalizes the Harris-Veech conditions for countable $\mathscr{X}$.

Girsanov showed that under an absolutely continuous change in probability measure a Wiener process is transformed into the sum of a Wiener process and a second process with sample functions which are absolutely continuous. This result has a natural generalization in the context of local martingales. This generalization is derived in this paper, and some of its ramifications are examined. As a simple application, the likelihood ratio for a single-server queueing process with very general arrival and service characteristics is derived.

Let $\{X_n\}$ be a sequence of i.i.d. mean zero random variables. Let $S_n = X_1 + \cdots + X_n$. This paper is devoted to determining the conditions where-by $E\sup_{n\geqq 1}S_n/a_n < \infty$ and $E\sup_{n\geqq 1}|S_n|/a_n < \infty$ for quite general sequences of increasing constants $\{a_n\}$. For the sequences $\{a_n\}$ considered, we find it sufficient to examine whether or not $\lim_{n\rightarrow\infty} E(\sum^n_{k=1}X_k/a_k)^+ < \infty$. The existence of optimal extended-valued stopping rules with finite expected reward for sequences $\{S_n/a_n\}$ or $\{|S_n|/a_n\}$ is a by-product of our results. This generalizes results of D. L. Burkholder, Burgess Davis, R. F. Gundy, B. J. McCabe and L. A. Shepp, who treat the case $a_n = n$.

Let $(\mathscr{A}, \leqq)$ be a partially ordered set, $\{X_\alpha\}_{\alpha\in\mathscr{A}}$ a collection of i.i.d. random variables with mean zero, indexed by $\mathscr{A}$. Let $S_\beta = \sum_{\alpha\leqq\beta} X_\alpha, |\beta| = \operatorname{card} \{\alpha\in\mathscr{A}: \alpha \leqq \beta\}$. We study the a.s. convergence to zero of $Z_\beta = S_\beta/|\beta|$, when $|\beta| \mapsto \infty$. We first derive a Hajek-Renyi inequality for $K^r = \{(k_1, k_2, \cdots, k_r): k_i$ a positive integer$\}$. This is used to derive a sufficient condition for the convergence of $Z_\beta$ for a class of partially ordered sets including $K^r$. For many of these sets (and certain other sets as well) this condition is shown to be necessary. Finally a weaker sufficient condition is derived for a much larger class of sets, giving a theorem analogous to one of Hsu and Robbins for the linearly ordered case.

This paper studies Strong Laws of Large Numbers for Banach space-valued random variables which are subject to the Banach space analog of orthogonality called weak orthogonality.

Consider a dynamic programming problem with analytic state space $S$, analytic constraint set $A$, and semi-analytic reward function $r(x, P, y)$ for $(x, P)\in A$ and $y\in S$: namely, $\{r > a\}$ is an analytic set for all $a$. Let $Tf$ be the optimal reward in one move, with the modified reward function $r(x, P, y) + f(y)$. The optimal reward in $n$ moves is shown to be $T^n0$, a semi-analytic function on $S$. It is also shown that for any $n$ and positive $\varepsilon$, there is an $\varepsilon$-optimal strategy for the $n$-move game, measurable on the $\sigma$-field generated by the analytic sets.

Consider a dynamic programming problem with separable metric state space $S$, constraint set $A$, and reward function $r(x, P, y)$ for $(x, P)\in A$ and $y\in S$. Let $Tf$ be the optimal reward in one move, for the reward function $r(x, P, y) + f(y)$. Three results are proved. First, suppose $S$ is compact, $A$ closed, and $r$ upper semi-continuous; then $T^n0$ is upper semi-continuous, and there is an optimal Borel strategy for the $n$-move game. Second, suppose $S$ is compact, $A$ is an $F_\sigma$, and $\{r > a\}$ is an $F_\sigma$ for all $a$; then $\{T^n0 > a\}$ is an $F_\sigma$ for all $a$, and there is an $\varepsilon$-optimal Borel strategy for the $n$-move game. Third, suppose $A$ is open and $r$ is lower semi-continuous; then $T^n0$ is lower semi-continuous, and there is an $\varepsilon$-optimal Borel measurable strategy for the $n$-move game.

A real stochastic process $\{X(t): 0 \leqq t \leqq 1\}$, is called window-deterministic if the points $(t, X(t))$ on its graph belonging to a "window" $\{(t, x): 0 \leqq t \leqq 1, a < x < b\}$ stochastically determine all other points on the graph. Here it is shown that a large class of Gaussian processes with discontinuous sample functions has this property.

Let $\{\xi_n\}$ be a stationary Gaussian sequence with $E(\xi_0) = 0; E(\xi_0 \xi_n) = r_n$. If $n^\alpha r_n \rightarrow 0$ for some $\alpha > 1$, then Strassen's functional law of iterated logarithm applies to $\{\xi_n\}$.

Let $(\Omega, \mathscr{F}, P)$ be a probability space and let $\mathscr{F}_t, t \in R_+$, be an increasing family of sub-$\sigma$-fields of $\mathscr{F}$ such that $\mathscr{F}_0$ is trivial and $\mathscr{F} = V_t\mathscr{F}_t$. Let $\mathscr{M}^2$ be the family of all square-integrable martingales $m_t$ with $m_0 = 0$. Suppose that $L^2(\Omega, \mathscr{F}, P)$ is separable. Then there exists a finite or countable sequence in $\mathscr{M}^2, m_t^1, m_t^2, \cdots$, such that (i) the stable subspaces generated by $m_t^i, m_t^j$ are orthogonal for $i \neq j$; (ii) $\langle m^1\rangle \succ \langle m^2\rangle \succ\cdots$ where $\langle m^i\rangle$ is the nonnegative measure on the predictable $\sigma$-field on $\Omega \times R_+$ induced by the quadratic variation process $\langle m^i\rangle$ of $m^i$, and (iii) every $m$ in $\mathscr{M}^2$ has a representation $m_t = \sum_i \int^t_0 \phi_i(s) dm_s^i$ a.s. for some predictable integrands $\phi_i$. Furthermore, if $n_t^1, n_t^2, \cdots$ is another such sequence, then $\langle n^i\rangle \sim \langle m^i\rangle$ for all $i$.