If $\mathscr{A}$ is the tail, invariant, or symmetric field for discrete-time processes, or a field of the form $\mathscr{F}_{t+}$ for continuous-time processes, then no countably additive, regular, conditional distribution given $\mathscr{A}$ is proper. A notion of normal conditional distributions is given, and there always exist countably additive normal conditional distributions if $\mathscr{A}$ is a countably generated sub $\sigma$-field of a standard space. The study incidentally shows that the Borel-measurable axiom of choice is false. Classically interesting subfields $\mathscr{A}$ of $\mathscr{B}$ possess certain desirable properties which are the defining properties for $\mathscr{A}$ to be "regular" in $\mathscr{B}$.

We consider initial point processes $A_0$ on $Z^d$ where $A_0(x), x \in Z^d$ are independent and satisfy certain technical conditions. The particles initially present are assumed to be translated independently according to recurrent random walks. Various limit theorems are then proved involving $S_n(B)$--the total occupation time of $\mathbf{B}$ by time $n$, and $L_n(\mathbf{B})$--the number of distinct particles in $\mathscr{B}$ by time $n$.

Finkelstein's (1971) functional law of the iterated logarithm for empirical distributions is extended to cases where the empirical distribution is multiplied by a weight function, $w$. We let $X_1, X_2, \cdots$ be independent random variables each having the uniform distribution on $\lbrack 0, 1 \rbrack$, with $F_n$ the empirical df at stage $n$. The weight function $w$, defined on $\lbrack 0, 1 \rbrack$, is assumed to be bounded on interior intervals and to satisfy some smoothness conditions. Then convergence of the integral $\int^1_0 w^2(t)/\log \log(t^{-1}(1 - t)^{-1})dt$ is seen to be a necessary and sufficient condition for the sequence $\{U_n: n \geqq 3\}$, defined by $$U_n(t) = \frac{n^{\frac{1}{2}}w(t)(F_n(t) - t)}{(2 \log \log n)^{\frac{1}{2}}}$$ to be uniformly compact on a set of probability one, with set of limit points $$K_w = \{wf: f \in K\}$$. $K$ is the set set of absolutely continuous functions on $\lbrack 0, 1 \rbrack$ with $f(0) = 0 = f(1)$ and $$\int^1_0 \lbrack f'(t) \rbrack^2 dt \leqq 1.$$

The principal results of Resnick and Neuts (1970) and Resnick (1971) concerning limiting distributions for the maxima of a sequence of random variables defined on a Markov chain have been extended to denumerable Markov chains. These results apply a fortiori to Markov renewal processes. The method of proof is to show that limit distributions are independent of the initial distribution of the chain and then to apply known results for stationary processes.

The $GI/G/\infty$ queue is studied. For the stable case $(\nu = \text{expected service time} < \infty)$, necessary and sufficient conditions are given for the process to to have a legitimate regeneration point. In the unstable case $(\nu = \infty)$, several limit theorems are established. Let $X(t)$ equal the number of servers busy at time $t$. It is proven that when $\nu = \infty$, \begin{equation*}\tag{i}\frac{X(t)}{\lambda(t)} \Rightarrow 1\end{equation*} and \begin{equation*}\tag{ii}\frac{X(t) - \lambda(t)}{\sqrt{\lambda(t)}} \Rightarrow N(0, 1)\end{equation*} where $\lambda(t)$ is a deterministic function. ($\Rightarrow$ means convergence in distribution). A Poisson type limit result is also proved when the arrival of a customer is a rare event.

If $B_n$ denotes the time of the first birth in the $n$th generation of an age-dependent branching process of Crump-Mode type, then under a weak condition there is a constant $\gamma$ such that $B_n/n \rightarrow \gamma$ as $n \rightarrow \infty$, almost surely on the event of ultimate survival. This strengthens a result of Hammersley, who proved convergence in probability for the more special Bellman-Harris process. The proof depends on a class of martingales which arise from a `collective marks' argument.

This paper provides a generalization of the classical Berry-Esseen theorem in two dimensions. For i.i.d. random variables $\eta_1, \eta_2, \cdots, \eta_r, \cdots$ and real numbers $a_1, a_2, \cdots, a_r, \cdots$ and $b_1, b_2, \cdots, b_r, \cdots$ with $E(\eta_r) = 0, E(\eta_r^2) = 1, |a_r| \leqq 1$ and $|b_r| \leqq 1$, let $A_n^2 = \sum^n_{r=1} a_r^2, B_n^2 = \sum^n_{r=1} b_r^2$ and $S_n = (\sum^n_{r=1} a_r \eta_r/A_n, \sum^n_{r=1} b_r \eta_r/B_n)$. The main result concerns the rate of convergence of the distribution function of $S_n$ to the corresponding normal distribution function without assuming the existence of third moments. As an application of this result a theorem of P. Erdos and A. C. Offord is generalized.

Let $Q_n$ be the distribution of the normalized sum of $n$ independent random vectors with values in $R^k$, and $\Phi$ the standard normal distribution in $R^k$. In this article the error $|\int f d(Q_n - \Phi)|$ is estimated (for essentially) all real-valued functions $f$ on $R^k$ which are integrable with respect to $Q_n$ when $s$th moments are finite, and for which the error may be expected to go to zero. When specialized to known examples, the (main) error bound provides precise rates of convergence.

This paper contains a general dependent extension of Doob's inequality for martingales, $E(\max_{i\leqq n} S_i^2) \leqq 4ES_n^2$. This inequality is then used to extend the martingale convergence theorem for $L_2$ bounded variables, and to prove strong laws under dependent assumptions. Strong and $\varphi$-mixing variables are shown to satisfy the conditions of these theorems and hence strong laws are proved as well for these.

Burkholder [3] gave a simple necessary and sufficient condition, in terms of concentration functions, for independent sequences of random variables to have the Stein property. Here we find sufficient conditions for the Stein property without assuming that the random variable sequence is independent. Our conditions are also in terms of concentration functions, but in our case they are conditional concentration functions which specialize to those used by Burkholder. For some of our results, the sequence of random variables may be quite arbitrary; however, we usually assume it to be a martingale difference sequence satisfying certain regularity conditions.

Let $M_k(0, T)$ denote the $k$th (factorial) moment of the number of zero crossings in time $T$ by a stationary Gaussian process. We present a necessary and sufficient condition for $M_k(0, T)$ to be finite. This condition is then applied to processes whose covariance functions $\rho(t)$ satisfy the local condition. $$\rho(t) = 1 - \frac{t^2}{2} + \frac{C|t|^3}{6} + o|t|^3$$ for $t$ near zero $(C > 0)$. In this case we show all the crossing moments $M_k(0, T)$ are finite. In the course of the proof of this result, we point out an error which vitiates the related work of Piterbarg (1968) and Mirosin (1971, 1973, 1974a, 1974b). We also find a counterexample to Piterbarg's results.

An explicit construction of invariant measures for a certain class of continuous state Markov processes is presented. A special version of these processes is of interest in the theory of representation of real numbers ($\beta$-expansions). Previous results of Renyi and Parry are generalized, and an open problem of Parry is resolved.

The tail behavior of a stable measure with respect to a seminorm is determined. Bounds are obtained for the measure of small spheres. The geometric structure of the support of stable measures with index $p > 1$ is described.

A simple derivation of a limiting distribution of Logan, Mallows, Rice and Shepp is found. This is the limiting distribution of a sequence of partial sums of independent, identically distributed random variables in the domain of attraction of a stable law, normalized by the partial sums of the absolute values of these variables.

A strong law of large numbers is shown for random sets taking values in the nonempty, compact subsets of $\mathbb{R}^n$.

Let $S_n, n = 1, 2, 3, \cdots$ denote the recurrent random walk formed by the partial sums of i.i.d. lattice random variables with mean zero and finite variance. Let $T_{\{x\}} = \min \lbrack n \geqq 1 \mid S_n = x \rbrack$ with $T \equiv T_{\{0\}}$. We obtain a local limit theorem for the random walk conditioned by the event $\lbrack T > n \rbrack$. This result is applied then to obtain an approximation for $P\lbrack T_{\{x\}} = n \rbrack$ and the asymptotic distribution of $T_{\{x\}}$ as $x$ approaches infinity.

Let $\mathscr{J}$ be a Boolean algebra of subsets of a state space $S$ and let $\succ$ be a binary comparative probability relation on $\mathscr{J}$ with $A \succ B$ interpreted as "$A$ is more probable than $B$." Axioms are given for $\succ$ on $\mathscr{J}$ which are sufficient for the existence of a finitely additive probability measure $P$ on $\mathscr{J}$ which has $P(A) > P(B)$ whenever $A \succ B$. The axioms consist of a necessary cancellation or additivity condition, a simple monotonicity axiom, an axiom for the preservation of $\succ$ under common deletions, and an Archimedean condition.

Given $N$ points distributed at random on $\lbrack 0, 1)$, let $p_n$ be the size of the smallest interval that contains $n$ points. Previous work finds $\operatorname{Pr}(p_n \leqq p)$ for all $n, N$, and rational $p$. The present note derives a new and considerably simplified formula for $\operatorname{Pr}(p_n \leqq p)$ for all $n, N$, and $p$.

We modify certain results obtained by I. A. Ibragimov concerning the remainder term in the Central Limit Theorem and the remainder term in the approximation of the distribution function for normed sums of i.i.d. rv's by a portion of the Chebyshev series and we then use these to complete some results appearing in a joint article by C. C. Heyde and the author (Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 21 255-268) (as well as in a similar article by F. N. Galstian (Theor. Probability Appl. 16 3, 528-533)).

Various asymptotically correct bounds on the uniform metric for distance between distribution functions in the central limit theorem for sums of independent and identically distributed random variables have previously been given. It is shown in the present paper that corresponding nonuniform bounds can be given for the difference between distribution functions. These results have much wider applicability, such as for obtaining probabilities of moderate deviation or for dealing with $L_p$ metrics, $1 \leqq p \leqq \infty$.