Let $\{\alpha_n\}$ be a sequence of independent, identically distributed random variables with $0 \leqq \alpha_n \leqq 1$ for all $n$. The random walk in a random environment on the integers is the sequence $\{X_n\}$ where $X_0 = 0$ and inductively $X_{n+1} = X_n + 1, (X_n - 1)$, with probability $\alpha_{X_n}, (1 - \alpha_{X_n})$. In this paper we consider limit theorems for the random walk in a random environment. We show that "randomizing the environment" in some sense "slows down" the random walk in Section One. The remaining sections are concerned with features of this "slowing down" in some simple models.

We propose a control problem in which we minimize the expected hitting time of a fixed state in an arbitrary Markov chains with countable state space. A Markovian optimal strategy exists in all cases, and the value of this strategy is the unique solution of a nonlinear equation involving the transition function of the Markov chain.

It is well known that a contraction multiplicative functional $\alpha_t, t \geqq 0$ on some Markov process with transition $P_t, t \geqq 0$, yields another Markov process whose semigroup $Q_t(x, A) = E_x(\alpha_t, X_t \in A)$ is subordinate to $P_t, t \geqq 0$. The second process results from the original one by adding a killing operation at a rate of $-d\alpha_t/\alpha_t$. This paper deals with expansion multiplicative functionals (satisfying $\alpha_t \geqq 1$ and $E_x(\alpha_t) < \infty)$. It is proved that such functionals yield a Markov process with creation and annihilation of mass. Relations to the original process are established. Finally the results are generalized to, so-called, conditionally monotone functionals.

The asymptotic behavior of the joint generating function of a critical age-dependent branching process is derived and used to obtain conditioned limit laws and a limiting diffusion.

In attempting to determine the growth properties of a branching process (b.p.), a standard method of attack is to look for the appropriate martingale. Here we show that for many b.p., this really corresponds to looking for the harmonic functions associated with the space-time process. As a particular application of the above we show that in the case of the classical Galton-Watson continuous time process with $m < \infty$ there exists constants $c(t)$ such that $Z_t/c(t)$ converges w.p. 1 to a nontrivial random variable.

If $F$ is an infinitely divisible distribution function without a Gaussian component whose Levy spectral measure $M$ is absolutely continuous and $M(\mathbb{R}^1\backslash\{0\}) = \infty$, then $F$ is shown to have an a.e. positive density over its support; this support of $F$ is always an interval of the form $(-\infty, \infty), (-\infty, a\rbrack$ or $\lbrack a, \infty)$. In addition, sufficient conditions are obtained for two infinitely divisible distribution functions without Gaussian components to be absolutely continuous with respect to each other, i.e., equivalent.

A relationship between the path structure of two real discrete time stochastic processes is deduced from inequalities between their transition functions. The approach is to define processes equivalent to the two on a common space so that pointwise inequalities are possible. An iterated logarithm type law for random walks is given as a particular application of the general method.

For any finitely additive probability measure to be disintegrable, that is, to be an average with respect to some marginal distribution of a system of finitely additive conditional probabilities, it suffices, and is plainly necessary, that the measure be conglomerative, that is, that there be a conditional expectation such that the expectation of no random variable can be negative if that random variable's conditional expectation given each of the marginal events is nonnegative. With respect to some margins, that is, partitions, there are finitely additive probability measures that are so far from being disintegrable that they cannot be approximated in the total variation norm by those that are. Those partitions which have this property are determined. Many partially defined conditional probabilities, and in particular, all disintegrations, or, equivalently, strategies, are restrictions of full conditional probabilities $Q = Q(A \mid B)$ defined for all pairs of events $A$ and $B$ with $B$ non-null.

Watch a martingale with uniformly bounded increments until it first crosses the horizontal line of height $a$. The sum of the conditional variances of the increments given the past, up to the crossing, is an intrinsic measure of the crossing time. Simple and fairly sharp upper and lower bounds are given for the Laplace transform of this crossing time, which show that the distribution is virtually the same as that for the crossing time of Brownian motion, even in the tail. The argument can be adapted to extend inequalities of Bernstein and Kolmogorov to the dependent case, proving the law of the iterated logarithm for martingales. The argument can also be adapted to prove Levy's central limit theorem for martingales. The results can be extended to martingales whose increments satisfy a growth condition.

An almost sure invariance principle due to Strassen for partial sums $\{S_n\}$ of martingale differences $\{X_n\}$ is sharpened. This result is then used to establish integral tests which characterize the asymptotic growth rates of $S_n$ and $M_n = \max_{1\leqq i\leqq n} |S_i|$. If, in addition, $\{X_n\}$ is a stationary ergodic sequence, then integral tests are established for nonrandom normalizers of $\{S_n\}$. Improving a decomposition due to Gordin for mixing sequences, integral tests are established for mixing sequences and Doeblin processes. In the independent case, the results obtained compare favorably with similar classical results due to Feller and strengthen a classical result due to Chung.

Some geometric properties of PD's are established, Kullback's $I$-divergence playing the role of squared Euclidean distance. The minimum discrimination information problem is viewed as that of projecting a PD onto a convex set of PD's and useful existence theorems for and characterizations of the minimizing PD are arrived at. A natural generalization of known iterative algorithms converging to the minimizing PD in special situations is given; even for those special cases, our convergence proof is more generally valid than those previously published. As corollaries of independent interest, generalizations of known results on the existence of PD's or nonnegative matrices of a certain form are obtained. The Lagrange multiplier technique is not used.

Formulas are given for the expectation and variance of the number of components for two definitions of random graphs. The results extend those of R. F. Ling (1973).

We prove an inequality for characteristic functions of lattice distributions, which can be used to give "explicit" remainder term estimates in local central limit theorems.

The extent $\mathscr{E}$ of a probability distribution is defined to be the range of its probability measure, thus, it represents the set of possible probabilities of events. The structure of $\mathscr{E}$ is examined for discrete distributions with (countably) infinite support. An application is considered.

The paper extends results of Levy and Taylor from one-parameter Brownian motion to the multiparameter case. The following theorem is proved. For $N$-parameter Brownian motion in $d$-space, the Hausdorff dimensions of the graph and range are a.s. $\min\{2N, N + d/2\}$ and $\min\{2N, d\}$ respectively.

Let $M_n = \max\{X_1, \cdots, X_n\}$ and $m_n(t) = (M_{\lbrack nt\rbrack} - a_n)/b_n(t \geqq 1/n)$, where the $\{X_i\}$ are independent rv's and $a_n$ and $b_n > 0$ are real constants. Suppose all the finite-dimensional laws of $m_n$ converge to those of a stochastic process $m = \{m(t): t > 0\}$. This paper is a study of the class of all such processes $m$.

Let $Y_n (n = 0, 1, \cdots)$ be a random variable and suppose that for suitably chosen constants $a_n$ and $b_n (b_n > 0)$ and each $t \in (0, \infty)$, the random variable $b_n Y_{\lbrack nt \rbrack} + a_n$ has a limit distribution function $G_t$. If $G_1$ is non-degenerate there are only two ways in which $G_t$ is related to $G_1$: there exist real constants $c$ and $\theta$ such that either $G_t(x) = G_1(c + t^\theta(x - c))$ for all $t > 0$ or else $G_t(x) = G_1(x + c \log t)$ for all $t > 0$. This result provides a very short derivation of the three types of extreme value limit distributions.

Let $S$ be the number of successes in $n$ independent Bernoulli trials, where $p_j$ is the probability of success on the $j$th trial. Let $\mathbf{p} = (p_1, p_2, \cdots, p_n)$, and for any integer $c, 0 \leqq c \leqq n$, let $H(c \mid \mathbf{p}) = P\{S \leqq c\}$. Let $\mathbf{p}^{(1)}$ be one possible choice of $\mathbf{p}$ for which $E(S) = \lambda$. For any $n \times n$ doubly stochastic matrix $\Pi$, let $\mathbf{p}^{(2)} = \mathbf{p}^{(1)}\Pi$. Then in the present paper it is shown that $H(c \mid \mathbf{p}^{(1)}) \leqq H(c \mid \mathbf{p}^{(2)})$ for $0 \leqq c \leqq \lbrack\lambda - 2\rbrack$, and $H(c \mid \mathbf{p}^{(1)}) \geqq H(c \mid \mathbf{p}^{(2)})$ for $\lbrack\lambda + 2\rbrack \leqq c \leqq n$. These results provide a refinement of inequalities for $H(c \mid \mathbf{p})$ obtained by Hoeffding [3]. Their derivation is achieved by applying consequences of the partial ordering of majorization.