This paper is mostly expository and is concerned with the connection between two dimensional Brownian motion and analytic functions provided by Levy's result that, if $Z_t, 0 \leqslant t < \infty$, is two dimensional Brownian motion, and if $f$ is analytic and not constant, then $f(Z_t), 0 \leqslant t < \infty$, is also two dimensional Brownian motion, perhaps moving at a variable speed. This can be used to study Brownian motion via analytic functions and, conversely, to treat analytic functions probabilistically. Recently several open problems in analytic function theory have been solved in this manner. We will present some of Doob's earlier work on the range and boundary values of analytic functions, the probabilistic theory of $H^p$ spaces due to Burkholder, Gundy and Silverstein, the author's results on conjugate function inequalities. and sketch probabilistic proofs of Picard's big and little theorems, and other theorems. There are some new results related to Hayman's generalization of Koebe's theorem.

A generalized problem of optimal stopping is defined, which has a solution in the class of quasi-stopping times, which generalize the classical stopping times. The optimal quasi-stopping times are fully characterized. By time reversal, on a Markov process, various results of Meyer, Mokobodzki and Rost are interpreted as results of existence of optimal quasi-stopping times for the reversed process. Finally, a game on quasi-stopping times is solved.

Using and refining a technique developed by O. Thorin, we prove: THEOREM. Let $f(x) = C\cdot x^{\beta - 1} h(x), x > 0$, be a probability density on $(0, \infty)$. Here $\beta > 0$ and $h$ is continuous and satisfies $h(0) = 1$. Assume that $h$ can be analytically continued to the whole complex plane cut along the negative real axis and assume that $h$ satisfies some other regularity assumptions. If $h$ is completely monotone on $(0, \infty)$ and if, for each fixed $u > 0$, the function $h(u\nu(t))h(u/\nu(t))$, where $\nu(t) = t + 1 + (t^2 + 2t)^{\frac{1}{2}}$, is completely monotone on $(0, \infty)$, then $f(x)$ is the density of a generalized gamma convolution and hence infinitely divisible. The theorem is applied to show the infinite divisibility of a rather large class of probability densities on $(0, \infty)$. In particular we show that a power with exponent of modulus $\geqslant 1$ of the ratio of two gamma distributed rv's has an infinitely divisible distribution.

A Hilbert space law of the iterated logarithm is proved which generalizes Kolmogorov's law for bounded random variables and which generalizes results of Teicher for unbounded random variables. The result for identically distributed random vectors is a consequence. The key idea is the requirement of the convergence of the average of the covariance operators.

Hampel's general qualitative definition of robustness of sequences of estimators on memoryless observation processes is generalized to stationary ergodic processes by substituting the generalized Ornstein (or $\bar{\phi}$) distance for the marginal Prohorov distance as the measure of "closeness" of observations. More general sequences of estimators are also allowed. The approach yields results analogous to those of Hampel for the more general case considered, often provides strict generalizations of Hampel's results, and in some cases yields simpler proofs.

Appropriate constant multiples of the function $t^{1/\alpha}$ are "the" "maximal local lower envelope" and "the" "minimal local upper envelope" for the sample functions of a strictly stable subordinator of index $\alpha$. The fact that the probability of extinction of a Galton-Watson process is less than one if the mean number of offspring is larger than one is used in the proofs.

We define a string of length $m$ to be a sequence $z_0, z_1,\cdots, z_m$ of points in $Z^2$ which are chosen so that $z_{i-1}$ and $z_i$ are adjacent points on the lattice. If we consider the set of all strings from $z_0 = (0, 0)$ to $z_m = (n, 0)$ then we can introduce a probability distribution by assuming that all the strings are equally likely. In this paper we will prove some results which describe the shape of the random string when $m/n\rightarrow \lambda \in (1, \infty)$. The main result states that if we let $(U_k, V_k)$ be the coordinate of $z_k$ then $(U_{\lbrack m\cdot \rbrack}/n, V_{\lbrack m\cdot \rbrack}/n^{\frac{1}{2}})$ converges weakly to $(\cdot, \rho^{\frac{1}{2}}W_0(\cdot))$ where $W_0$ is a Brownian bridge and $\rho = (\lambda^2 - 1)/2\lambda$. A second set of results describes $B_n = \sup\{U_k - U_l: 0 \leqslant k \leqslant l \leqslant m\}$, a quantity which measures the amount of backtracking in the string. We find that this is $0(\log n)$ and there is a sequence of constants $b_n$ so that $B_n - b_n$ approaches a double exponential distribution. The results described above were motivated by, and are related to, results of Abraham and Reed, and Gallovotti on the shape of the interface profile in the two dimensional Ising model.

The well-known concept of conjugate slowly varying functions is specialized to the subclass $\Pi$ of the slowly varying functions. The concept is then used to connect convergence of certain increasing stochastic processes (suitably normalized) with convergence of their inverses.

Let $\{X_n, n \geqslant 1\}$ be a sequence of independent random variables uniformly distributed on the unit interval. Put $X^\ast_n = \inf(X_1, X_2,\cdots, X_n)$ and $S_n = X^\ast_1 + X^\ast_2 + \cdots + X^\ast_n, n \geqslant 2, S_1 = 0$. The aim of this note is to give an almost sure invariance principle for $S_n$. Next we extend the given results to the case when $X_n, n \geqslant 1$, are not uniformly distributed but bounded, and moreover, to sums $\hat{S}_n = X^{(m)}_m + X^{(m)}_{m+1} +\cdots + X^{(m)}_n$, where $X^{(m)}_j$ is the $m$th order statistic of $(X_1, X_2,\cdots, X_j)$.

Let $(X, \mathscr{a}, P)$ be a probability space, $Y$ a complete separable metric space, $Z$ a separable metric space, and $s: X\rightarrow Y, t: X\rightarrow Z$ Borel measurable functions. Then the weak limit of $P\{s \in B, t \in C\}/P\{t \in C\}$ for $C\downarrow\{z\}$ exists for $P-\mathrm{a.a.} z \in Z$, and is a regular conditional distribution of $s$, given $t$.

An inequality for certain random sequences more general than martingales or nonnegative submartingales is proved. Three special cases are deduced, one of which generalizes and refines a result of Austin. As an application of the inequality, the special cases are used to give new proofs of Burkholder's $L \log L$ and $L_p$ (for $1 < p \leqslant 2$) inequalities for the square function of a martingale or a nonnegative submartingale.

A construction is given for a martingale having a given right continuous strictly positive submartingale as its absolute value. The existence of a martingale whose absolute value has the same distribution as the given submartingale was shown by Gilat.

Let $\{S_n; n \geqslant 1\}$ denote the partial sums of i.i.d. random variables and let $\{n_k; k \geqslant 1\}$ be a (strictly) increasing subsequence of the positive integers. We determine necessary and sufficient conditions for $E \sup_k|S_{n_k}/n_k| < \infty$.

Take $n$ points at random on a circle of unit circumference and order them clockwise. Let $S^{(m)}_0,\cdots, S^{(m)}_{n-1}$ be the $m$th order spacings, i.e., the clockwise arc-lengths between every pair of points with $m - 1$ points between. Ordinary spacings correspond to the case $m = 1$. A central limit theorem is proved for $Z_n = \sum^{n-1}_{k=0}h(nS_k,\cdots, nS_{k+m-1})$, where $h$ is a given function. Using this, asymptotic distributions of central order statistics and sums of the logarithms of $m$th order spacings are derived.

Using a technique developed by Chen, we obtain an analogue of Komlos' theorem and a simple proof of a strong law of large numbers for strategies.

An exponential limit distribution is obtained for stopping times associated with partial sums of independent, identically distributed random variables whose distribution function is slowly varying at infinity. It is also demonstrated that a generalized law of the iterated logarithm cannot obtain in such a case.

Earlier papers have shown how to convert competing risk models involving dependent random variables into models involving only independent random variables, while simultaneously preserving the distribution of the minimum and the probabilities of the various failure patterns. In the present paper, we consider a sequence of such conversions occurring at successive points in chronological time in which the independent random variables are becoming stochastically larger. We obtain results which essentially demonstrate that the limiting distributions in the sequence of dependent models "correctly" correspond to the limiting distributions in the sequence of independent models. These results have applications in reliability growth models and in biomedical competing risk models in which the competing risks are increasing with age; in these models dependency is permitted among the random variables.

Basterfield showed that if $X \in L \log L$ and $\{\mathscr{F}_n\}$ form a sequence of independent $\sigma$-fields, then $E(X\mid\mathscr{F}_n)\rightarrow EX$ a.s. His proof uses the theory of Orlicz spaces. We generalize Basterfield's theorem to the case of Markov-dependent $\sigma$-fields and also weaken the restrictions on $X$. Our approach is different from Basterfield's in that it is martingale-theoretic.

The limiting distribution of $\max_{1\leqslant k \leqslant n}S_k/k^{\frac{1}{2}}$ is derived via embedding. The theorem also applies to partial sums of certain dependent rv's. Thus the proof of the Darling and Erdos result is brought into line with recent literature; moreover, the scope of its applicability is greatly increased in regard both to dependence and moment assumptions.