Let $X_1, X_2, \cdots$ be i.i.d. nondegenerate mean zero random variables with common distribution function $F$ such that $EX_1^+ \log^+ X_1 < \infty$. Let $S_n = X_1 + \cdots + X_n$ and $T_\infty$ be the collection of randomized extended-valued stopping rules. For $t\in T_\infty$, define $E(S_t/t) = E(S_t/t)1_{\{t < \infty\}}$. There exists $\tau\in T_\infty$ such that $E(S_\tau/\tau) = \sup_{t\in T_\infty} E(S_t/t) < \infty$. We wish to determine whether or not $P(\tau = \infty) > 0$. Let $b^{-1}(x) = 1/(\int^\infty_x (y/x)\log (y/x) dF(y))$. We find that $P(X_n \geqq b(n) \mathrm{i.o.}) = 1 \Rightarrow P(\tau < \infty) = 1$. As a corollary, if $E(X_1^+)^\alpha < \infty$ for some $\alpha > 1, P(\tau < \infty) = 1$. Conversely, if $P(|X_n| \geqq b(n) \mathrm{i.o.}) = 0$, then $P(\tau = \infty) > 0$. Moreover, if $g(n)\nearrow$ and $\sum^\infty_{n=1} g(2^n)^{-1} < \infty$ then $P(|X_n| \geqq b(n)g(n)\mathrm{i.o.}) = 1 \Rightarrow P(\tau < \infty) = 1$. Examples satisfying these latter conditions are given. An outgrowth of this work is that for any i.i.d. sequence $\{X_n\}$ of mean zero random variables and $c < \frac{1}{4}, P(S_n \geqq cE|S_n|\mathrm{i.o.}) = 1$. The importance of this result stems from the fact that we may also have $S_n(\log n)^\varepsilon/n \rightarrow -\infty$ in probability (see [15]). In order to be completely rigorous, a section was included which provides a useful characterization of the general form of the class of optimal rules for sequential decision problems.

The coefficient de parente $\sigma(x, y)$ of Malecot, which measures the genealogical relatedness of individuals $x$ and $y$, is derived as a correlation coefficient of certain genotype scores. An explicit formula is given for $\sigma(x, y)$ involving path sums in the ancestral charts of $x$ and $y$. The expected value of $\sigma(x, y)$ is worked out for certain random population models.

On the intrinsic time-scale, a martingale with uniformly bounded increments escapes from a strip at the same asymptotic rate as Brownian motion.

If $X_i$ are i.i.d. and have zero mean and arbitrary finite variance the limiting probability distribution of $S_n(2) = (\sum^n_{i=1} X_i)/(\sum^n_{j=1} X_j^2)^{\frac{1}{2}}$ as $n \rightarrow \infty$ has density $f(t) = (2\pi)^{- \frac{1}{2}}\exp (-t^2/2)$ by the central limit theorem and the law of large numbers. If the tails of $X_i$ are sufficiently smooth and satisfy $P(X_i > t) \sim rt^{-\alpha}$ and $P(X_i < -t) \sim lt^{-\alpha}$ as $t\rightarrow\infty$, where $0 < \alpha < 2, r > 0, l > 0, S_n(2)$ still has a limiting distribution $F$ even though $X_i$ has infinite variance. The density $f$ of $F$ depends on $\alpha$ as well as on $r/l$. We also study the limiting distribution of the more general $S_n(p) = (\sum^n_{i=1} X_i)/(\sum^n_{j=1} |X_j|^p)^{1/p}$ where $X_i$ are i.i.d. and in the domain of a stable law $G$ with tails as above. In the cases $p = 2$ (see (4.21)) and $p = 1$ (see (3.7)) we obtain exact, computable formulas for $f(t) = f(t,\alpha, r/l)$, and give graphs of $f$ for a number of values of $\alpha$ and $r/l$. For $p = 2$, we find that $f$ is always symmetric about zero on $(-1, 1)$, even though $f$ is symmetric on $(-\infty, \infty)$ only when $r = l$.

In this paper, we study weighted sums $\sum^n_{i=1} c_{n-i} X_i$ of i.i.d. zero-mean random variables $X_1, X_2, \cdots$, under the condition that the sequence $(c_n)$ is square summable. It is proved that such weighted sums are, with probability 1, of smaller order than $n^{1/\alpha}$ (respectively $\log n$, etc.) $\operatorname{iff} E|X_1|^\alpha < \infty$ (respectively $Ee^{t|X_1|} < \infty$ for all $t < \infty$, etc.). Certain analogs of the law of the iterated logarithm for such weighted sums are also obtained.

In this paper, we are interested in moving averages of the type $\int^t_0 f(t - s) dX(s)$, where $X(t)$ is a Wiener process and $\int^\infty_0 f^2(t) dt < \infty$. By a suitable choice of the weighting function $f$, such processes can be used to detect a change in the drift of $X(t)$. First passage times of these moving-average processes and more general Gaussian processes are studied. Limit theorems for Gaussian processes and Gaussian sequences which include these moving-average processes and their discrete-time analogs as special cases are also proved.

Let $\{X_n\}$ be a sequence of independent random variables none of which are degenerate and define for $y \geqq 0, F(y) = \sup_k P\lbrack |X_k| \geqq y \rbrack$ and $G(y) = \sup_{j \neq k} P\lbrack |X_j X_k| \geqq y \rbrack$. Relationships between the rate of convergence of $F$ and $G$ to zero are investigated. Set $Q_N = \sum_{j, k}a_{jk,N} X_jX_k$ for $N = 1, 2, \cdots$. If the $X$'s are symmetric then it is shown that $Q_N$ converges to zero in probability for a large class of weights $\{a_{jk,N}\}$ if and only if $\lim_{y\rightarrow\infty} yG(y) = 0$. Convergence results are also given for the case when the random variables are not symmetric.

For processes $X(t)$ with stable components we calculate $\dim X(E)$ in terms of $\dim E$, where $E$ is a fixed Borel subset of [0, 1] of known Hausdorff-Besicovitch dimension, $\dim E$. Our results extend the earlier ones of Blumenthal and Getoor in the stable case.

We show that "hitting times" associated with stationary processes have an absolutely continuous distribution, except, possibly, for an atom at the origin. The density is then identified in several special cases.

An age dependent subcritical branching process is constructed for which the conditioned limits $$\lim_{t\rightarrow\infty} P\{Z(t) = k \mid Z(t) > 0\}\quad k \geqq 1,$$ do not exist.

Some theorems are obtained relating the asymptotic behavior of a distribution function with the local behavior of its characteristic function.

By embedding partial sum processes into Brownian motion, it is well known that the deMoivre-Laplace central limit theorem is a consequence of the strong law of large numbers. It is the purpose here to show that the embedding technique can be used to establish both the degenerate convergence criterion and the normal convergence criterion for triangular arrays of uniformly asymptotically negligible random variables.

It is shown that a theorem of Bilingsley ((1968) Convergence of Probability Measures, Wiley; Theorem 22.1) about weak convergence of empirical processes of $\varphi$-mixing sequences also holds for a class of strong-mixing sequences.

Exact, approximate, asymptotic, and computational formulas are derived for the expected number of components of any given size in a random linear graph. A theorem generalizes some asymptotic results of Austin, Fagen, Penney, and Riordan.