We give a simple representation of two-parameter martingales in terms of a stochastic integral. This representation leads to the idea of the partial derivate of a martingale and to a generalization of the stochastic Green's theorem of the authors. Green's formula in this generalized form gives us a new and simpler proof of the fact that the derivative of a holomorphic process is holomorphic.

Let $S$ be a countable set and let $\xi_t$ be a Markov process on the subsets of $S$. Harris has given criteria for the existence of a dual process $\xi_t^\ast$ on the finite subsets of $S$. By extending Harris's notion of duality the class of $\xi_t$ which have dual processes is enlarged. The dual processes are then used to study the ergodic behavior of $\xi_t$. Also treated is a class of $\xi_t$ which have growing dual processes.

Let $S$ be a countable set and let $Q$ be a stochastic matrix on $S \times S$. An entrance law for $Q$ is a collection $\mathbf{\mu} = \{\mu_n\}_{n\in\mathbb{Z}}$ of probability measures on $S$ such that $\mu_nQ = \mu_{n+1}$ for all $n\in\mathbb{Z}$. There is a natural correspondence between entrance laws and Markov chains $\xi_n$ with stationary transition probabilities $Q$ and time parameter set $\mathbb{Z}$. The set $\mathscr{L}(Q)$ of entrance laws is examined in the discrete and continuous time setting. Criteria are given which insure the existence of nontrivial entrance laws.

The following problem is solved: when are we obliged to walk upstairs in a building supplied with two lifts (or elevators), if repairs to the lifts are quickly carried out? Sufficient conditions are given on the distributions of operating- and down-times under which the times when both lifts are inoperative are asymptotically distributed as a Poisson process.

Stone has decomposed the renewal measure of a probability distribution $F$ into two parts: a finite component with the same tail behaviour as that of $F$ and an absolutely continuous one which is nearly stationary at infinity. Our theorem asserts the uniformity of this decomposition.

After a study of the process $Y$, obtained from a right process $X$ by time-changing it with respect to a continuous additive functional $A$, we relate the jumps of $Y$ in $\Phi = \operatorname{sup} A$ to the excursions of $X$ out of $\Phi$ and to the jumps of $X$ inside $\Phi$.

For Brownian motion with constant drift, when and where the first exit from $(-b, b)$ occur are independent random variables.

The total variation of a simple, symmetric random walk with absorbing barrier at zero, is stochastically larger than the total variation of any other nonnegative, integer-valued supermartingale with the same initial position. This strengthens a result of David Freedman on the optimality of timid play for maximizing the time to bankruptcy in certain gambling situations.

Let $A_1, A_2, \cdots, A_n$ be events on a probability space. Let $S_{k,n}$ be the $k$th binomial moment of the number $m_n$ of those $A$'s which occur. An estimate on the distribution $y_t = P(m_n \geqq t)$ by a linear combination of $S_{1,n}, S_{2,n}, \cdots, S_{n,n}$ is called a Bonferroni inequality. We present for proving Bonferroni inequalities a method which makes use of the following two facts: the sequence $y_t$ is decreasing and $S_{k,n}$ is a linear combination of the $y_t$. By this method, we significantly simplify a recent proof for the sharpest possible lower bound on $y_1$ in terms of $S_{1,n}$ and $S_{2,n}$. In addition, we obtain an extension of known bounds on $y_t$ in the spirit of a recent extension of the method of inclusion and exclusion.

Using the representation theorem and inversion formula for Stieltjes transforms, we give a simple proof of the infinite divisibility of the student $t$-distribution for all degrees of freedom by showing that $x^{-\frac{1}{2}}K_\nu(x^{\frac{1}{2}})/K_{\nu+1}(x^{\frac{1}{2}})$ is completely monotonic for $\nu \geqq -1$. Our approach proves the stronger and new result, that $x^{-\frac{1}{2}}K_\nu (x^{\frac{1}{2}}) /K_{\nu+1}(x^{\frac{1}{2}})$ is a completely monotonic function of $x$ for all real $\nu$. We also derive a new integral representation.

Let $X_1, X_2, \cdots$ be independent, identically distributed, nondegenerate random variables, let $w_k$ be a sequence of positive numbers and for $n = 1,2, \cdots$ let $S_n = \sum^n_{k=1} w_kX_k$ and $W_n = \sum^n_{k=1} w_k$. The weak (strong) law is said to hold for $\{X_k, w_k\}$ if and only if $S_n/W_n$ converges in probability (almost surely) to a constant. Jamison, Orey and Pruitt (1965) (Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 40-44) studied conditions related to these laws of large numbers. In considering the strong law, only distributions with finite first moments are discussed. However, Theorem 2 of this paper shows that a sequence of random variables and a sequence of weights can be chosen so that the strong law holds and so that the random variables have arbitrarily heavy tails. This result also answers some interesting questions concerning the weak law.

If $(u_n)^\infty_{n=0}$ is a null-recurrent renewal sequence, we prove that there exist two null-recurrent renewal sequences $(\nu_n)^\infty_{n=0}$ and $(w_n)^\infty_{n=0}$ such that $(u_n\nu_n)^\infty_{n=0}$ is null-recurrent and $(u_n w_n)^\infty_{n=0}$ is transient.

Let $X_n$ be a sequence of i.i.d. random variables with mean 0 and variance 1. Let $S_n^\ast = n^{-\frac{1}{2}} \sum^n_{\nu=1} X_\nu$. We investigate in this paper the convergence order in conditioned central limit theorems, that is, the convergence order of $\sup_{t\in\mathbb{R}}|P(S_n^\ast < t|B) - \phi(t)|$.

The joint Laplace transform of $T$ and $X(T)$ is derived where $X(\bullet)$ is a time homogeneous diffusion process and $T$ is the first time the process falls a specified amount below its current maximum. This generalizes the work of Taylor. The distribution of the maximum at $T$ is shown to be exponential for Brownian motion. Formulas for more general stopping times based on the current maximum are given.

The function $\varphi(h) = h/|\log h|$ is shown to be an exact Hausdorff measure function for the range of all strictly asymmetric Cauchy processes in $R^k, k \geqq 2$. The same function is also shown to correctly measure the graph of any strictly asymmetric Cauchy process.

In an earlier paper, the author proves an invariance principle for mixingales, a generalization of the concepts of mixing sequences and martingale differences, under the condition that the variance of the sum of $n$ random variables is asymptotic to $\sigma^2n$ where $\sigma^2 > 0$. In this note we relax further the required degree of stationarity, requiring only that the squared variables properly normalized form a uniformly integrable family, and the partial sums have variances consistent with the Wiener process.

Some conditions are introduced which imply pointwise convergence theorems for increasing sequences of orthogonal projections on $L^2(\mu), \mu$ finite, as well as a pointwise ergodic theorem for self-adjoint and unitary contractions. These results generalize to the case of nonpositive operators some theorems of E. M. Stein.

The expected rank of the individual selected in the secretary problem can be kept bounded as the number of individuals becomes infinite even if we are not permitted to remember more than one individual at a time. The least upper bound of these expected ranks is derived: it is approximately 7.4.

In a recent paper by J. Gianini and S. M. Samuels an "infinite secretary problem" was formulated: an infinite, countable sequence of rankable individuals (rank 1 = best) arrive at times which are independent and uniformly distributed on [0, 1]. As they arrive, only their relative ranks with respect to their predecessors can be observed. Given an increasing cost function $q(\bullet)$, let $\nu$ be the minimum, among all stopping rules, of the mean of the function $q$ of the actual rank of the individual chosen. Let $\nu(n)$ be the corresponding minimum for a finite secretary problem with $n$ individuals. Then $\lim \nu(n) = \nu$.

If $\{S_n\}$ is a genuinely $d$-dimensional random walk and $d \geqq 3$, then with probability 1, $n^{-\alpha}|S_n| \rightarrow \infty$ as $n \rightarrow \infty$ for every $\alpha < \frac{1}{2}$. This follows from a recent result of H. Kesten. In this paper we show that, under certain conditions, there is a constant $\alpha_0$ depending on the walk, but $\frac{1}{2} - 1/d \leqq \alpha_0 < \frac{1}{2}$, and a deterministic sequence of vectors $\{\nu_n\}$ such that $\lim \inf_n n^{-\alpha}|S_n - \nu_n| = 0$ with probability 1 for every $\alpha \geqq \alpha_0$. In discrete time this phenomenon cannot occur for any $\alpha < \frac{1}{2} - 1/d$; in continuous time it can occur for any $\alpha > 0$.