Necessary and sufficient conditions are given for boundedness of $\sup_n \|\sum^n_{k=1} (P^k(x, \bullet) - P^k(y, \bullet))\|$ and $\sup_n \|\sum^n_{k=1} (P^k(x, \bullet) - \pi\|$, where the norm is total variation and where $\pi$ is an invariant probability measure. Also conditions for convergence of $\sum^\infty_{k=1} (P^k(x, \bullet) - \pi)$ in norm are given. These results require the study of certain "small sets." Two new types are introduced: uniform sets and strongly uniform sets, and these are related to the sets introduced by Harris in his study of the existence of $\sigma$-finite invariant measure.

Let $\{X_t, t \geqq 0\}$ be a process with stationary independent increments taking values in $d$-dimensional Euclidean space. Let $S$ be a set in $R^d$, and let $T = \inf\{t > 0: X_t \not\in S\}$. For a reasonably wide class of processes and sets $S$, criteria are given for deciding when $P\{X_T \in B\} > 0$ and when $P\{X_T \in B\} = 0$, where $B \subset \partial S$.

We determine $E\lbrack \exp (\alpha X(T) - \beta T) \rbrack$ where $X(t)$ is a Brownian motion having arbitrary drift and variance and $T$ is the first time the process drops a specified amount below its maximum to date. From this result, the moments of $X(T)$ and $T$ and some asymptotic distributions may be found. Applications in process control and financial management are mentioned.

We establish integral tests for upper and lower functions for Brownian motion in Banach spaces of infinite-dimension as well as some related results for Brownian motion in $\mathbb{R}^d$.

We study stochastic integrals of the form $Y(t) = \int^t_0 V d X, t \geqq 0$, where $X$ is a process with stationary independent increments while $V$ is an adapted previsible process, thus continuing the work of Ito and Millar. In the case of vanishing Brownian component, we obtain conditions for existence which are considerably weaker than the classical requirement that $V^2$ be a.s. integrable. We also examine the asymptotic behavior of $Y(t)$ for large and small $t$, and we consider the variation with respect to suitable functions $f$. The latter leads us to investigate nonlinear integrals of the form $\int f(V dX)$. The whole work is based on extensions of two general martingale-type inequalities, due to Esseen and von Bahr and to Dubins and Savage respectively, and on a super-martingale which was discovered and explored in a special case by Dubins and Freedman.

Convergence in mean of $N^{-1} \sum^N_{k=1} X_{t_k}$ is studied for stationary processes classified according to parameter space and type of spectral measure.

Let $\xi_1, \xi_2, \cdots$ be independent random variables. The distribution of $\max (0, \xi_1, \xi_1 + \xi_2, \cdots, \xi_1 + \cdots + \xi_n)$ is investigated by means of a method based on the construction of certain events with easily determined proabilities. These yield a new formula for the distribution of the maximum which is sometimes more useful than that given in literature.

Given $(X_t)_{t\geqq1}$, independent random variables on some measurable space $(I, \mathscr{B})$ with the same distribution $m$, and a positive function $f$ of $L^1(m)$ with $\|f\|_1 = 1$, this paper studies how to build a stopping time $T$ with respect to the $\sigma$-fields $\mathscr{F}_t$ generated $X_1, X_2, \cdots, X_t$, such that the distribution of $X_T$ in $(I, \mathscr{B})$ is exactly $f dm$.

Let $(X_n, \mathscr{F}_n, n \geqq 1)$ be a martingale difference sequence with $E(X_n^2 \mid \mathscr{F}_{n-1}) = 1$ a.s. This paper presents iterated logarithm results involving $\lim \sup_{n\rightarrow\infty} \sum^n_{m=1} f(m/n)X_m/(2n \log \log n)^{\frac{1}{2}}$, where $f$ is a continuous function on [0, 1]. For example, it is shown that the above limit superior equals the $L_2$-norm of $f$ if the $X_n$'s are uniformly bounded and $f$ is a power series with radius in excess of one. These results generalize (and correct the proof of) a previous theorem due to the author. A generalization of the strong law of large numbers is also established.

Ornstein's $\bar{d}$ distance between finite alphabet discrete-time random processes is generalized in a natural way to discrete-time random processes having separable metric spaces for alphabets. As an application, several new results are obtained on the information theoretic problem of source coding with a fidelity criterion (information transmission at rates below capacity) when the source statistics are inaccurately or incompletely known. Two examples of evaluation and bounding of the process distance are presented: (i) the $\bar{d}$ distance between two binary Bernoulli shifts, and (ii) the process distance between two stationary Gaussian time series with an alphabet metric $|x - y|$.

Cramer's theorem, that a normal distribution function $(\operatorname{df})$ has only normal components, is extended to a case where the components are allowed to be from a subclass $(B_1)$ of the functions of bounded variation other than the class of df's. One feature of $B_1$ is that it contains more of the df's than the classes for which previous similar extensions have been made; in particular it contains the Poisson df's so that a first extension of Raikov's theorem, that a Poisson df has only Poisson components, in the same direction, is also given.

In this paper, it is shown that a random variable $Z$ having an infinitely divisible distribution with a finite left extremity can be decomposed in distribution as the sum of two random variables such that, given one, the other has an infinitely divisible distribution of prescribed type. Some extensions of this result when the left extremity is infinite are also given, and a new decomposition of a normal variable is exhibited.

Let $p(t), t \in (0, \infty)$ be the standard $p$-function of a regenerative phenomenon as defined in Kingman's theory. Let $p(1) = M$ and $\min \{p(t), 0 \leqq t \leqq 1\} = m$. Griffeath (1973) has derived a new upper bound for $M$ for given $m$ by using the Kingman inequalities of order $\leqq 3$. Here Griffeath's result is generalized by using the Kingman equalities of order $\leqq n$. Further taking limits as $n \rightarrow \infty$ a new upper bound is obtained which is uniformly strictly superior to the present known upper bound. Thus a part of the uncharted region in the $M - m$ diagram becomes charted by being shown inaccessible. This gives also an improved upper bound for the constant $\nu_0$.

The exponential distribution is often characterized as the only distribution with lack of memory. This note points out a stronger result: the exponential is the only distribution that is occasionally forgetful.

Let $(V_\lambda)$ be a sub-Markovian resolvent of kernels $V_\lambda$ on a measurable space $(E, \mathscr{E})$. Assume that $V = \lim_{\lambda \downarrow 0}V_\lambda$ is a proper kernel. The proper kernels $V$ on $(E, \mathscr{E})$ that are of the form $V = \lim_{\lambda \downarrow 0}V_\lambda, (V_\lambda)$ a sub-Markovian resolvent of kernels on $(E, \mathscr{E})$, are proved to be precisely those proper kernels $V$ which satisfy the complete maximum principle and for which the following condition holds: there exists an increasing sequence $(A_n) \subset \mathscr{E}$ with $\mathbf{\bigcup}_n A_n = E$ such that (i) $V1_{A_n} < \infty$ for all $n$; and (ii) if $f \in \mathscr{E}^+$ and $Vf < \infty$ then $\inf_nR_{\mathscr{C} A_n} Vf < \infty$, where $R_Bu = \inf \{v \text{supermedian} \mid u \geqq v \text{on} B\}$.

General diffusion approximation theorems are established for sequences of non-Markovian processes. These theorems cover certain genetic models previously considered by Watterson. It follows that Watterson's conclusions concerning these models are correct, even though there is a gap in his proof.

In this paper the following two theorems are shown: if $U, V$ are Burkholder type operators on martingales and if the inequality $E\lbrack U(X) \rbrack \leqq c \cdot E\lbrack V(X) \rbrack$ holds for every martingale $X$, then the inequality $E\lbrack F \circ U(X) \rbrack \leqq C \cdot E\lbrack F \circ V(X) \rbrack$ holds, for $F$ concave if $V$ is "predictable," for $F$ convex if $U$ is "predictable."

For $X_1, \cdots, X_n$ independent real valued random variables and for $\alpha \in \lbrack 0, 1 \rbrack$, let $F_j(x) = \alpha P\lbrack X_j < x \rbrack + (1 - \alpha)P\lbrack X_j \leqq x \rbrack$ and $Y_j(x) = \alpha I_{\lbrack X_j < x \rbrack} + (1 - \alpha) I_{\lbrack X_j \leqq x \rbrack}$, where $I_A$ is the indicator function of the set $A$. For numbers $w_1, w_2, \cdots, w_n$, let $D_n = \sup_{x, \alpha} \max_{N \leqq n}|\sum^N_1 w_j(Y_j(x) - F_j(x))|$. We will obtain an exponential bound for $P\lbrack D_n \geqq a \rbrack$ and a rate for almost sure convergence of $D_n$. When $w_j \equiv 1$ the bound and the rate become, respectively, $4a \exp \{-2((a^2/n) - 1)\}$ and $O((n \log n)^{\frac{1}{2}})$.

This article examines the properties of conditional medians, especially vis-a-vis those of conditional expectations.

Let $r_n$ be the probability that a recurrent random walk on a countable Abelian group fails to return to the origin in the first $n$ steps. For two-dimensional walk, Kesten and Spitzer have shown that $r_n$ is slowly varying. I.e. $\lim_{n\rightarrow\infty} r_{2n}/r_n = 1$. We strengthen this result and show that for any countable Abelian group of rank 2, $r_n$ is super slowly varying in the sense that $\lim_{n\rightarrow\infty} r_{\lbrack nr_n \rbrack}/r_n = 1$. We use the superslow variation of $r_n$ to obtain the limit law for the number of returns to the origin for all recurrent random walks on these groups.