Williams and Bjerknes introduced in 1972 a stochastic model for the spread of cancer cells; independently, this model has since surfaced within the field of interacting particle systems as the biased voter model. Cells, normal and abnormal (cancerous), are situated on a planar lattice. With each cellular division, one daughter stays put, while the other usurps the position of a neighbor; abnormal cells reproduce at a faster rate than normal cells. We treat here the long-term behavior of this system. In particular, we show that, provided it lives forever, the tumour will eventually contain a ball of linearly expanding radius. This also demonstrates the ergodicity of the interacting particle system, the coalescing random walk with nearest neighbor births. Our techniques include the use of dual processes, and of different numerical computations involving the use of imbedded processes.

Let $C_p$ be the limiting shape of Richardson's growth model with parameter $p \in (0, 1\rbrack$. Our main result is that if $p$ is sufficiently close to one, then $C_p$ has a flat edge. This means that $\partial C_p \cap \{x \in R^2:x_1 + x_2 = 1\}$ is a nondegenerate interval. The value of $p$ at which this first occurs is shown to be equal to the critical probability for a related contact process. For $p < 1$, we show that $C_p$ is not the full diamond $\{x \in R^2:\|x\| = |x_1| + |x_2| \leq 1\}$. We also show that $C_p$ is a continuous function of $p$, and that when properly rescaled, $C_p$ converges as $p \rightarrow 0$ to the limiting shape for exponential site percolation.

Let $X$ be a real valued random variable with probability distribution function $F(x)$ and characteristic function $c(t)$. Let $F_n(x)$ be the $n$th empirical distribution function associated with $X$ and $c_n(t)$ the characteristic function of $F_n(x)$. Necessary and sufficient conditions are obtained for the weak convergence of $\sqrt{n}\lbrack c_n(t) - c(t)\rbrack$ on the space of continuous complex valued functions on $\lbrack -1/2, 1/2\rbrack$.

In this paper we show that a necessary and sufficient condition on a Banach space $B$ for the validity of the accompanying laws theorem is that $c_0$ is not finitely representable in $B$ or, equivalently, that $B$ is of cotype $q$ for some $q > 0$. The proof is based on a result of Maurey and Pisier on the geometry of these spaces and on a theorem about approximation in $L_p$ of Banach valued triangular arrays by finite dimensional ones.

Let $(E, B)$ be a measurable vector space and $q$ be a measurable seminorm on $E$. Suppose that $(\mu_t)_{t > 0}$ is a $q$-continuous convolution semigroup of probability measures on $(E, B)$. It is proved that there exists a right-continuous nonincreasing function $\theta$ such that $\lim_{t \rightarrow 0+} (1/t)\cdot \mu_t\{x: q(x) > s\} = \theta(s)$ for every $s > 0$ at which $\theta$ is continuous. If $\mu_t, t > 0$, are Gaussian, then $\theta \equiv 0$; if there exists a measurable linear functional $f$ such that $f(\cdot)$ is not Gaussian (with respect to $\mu_1$) and $q \geqslant |f|$ then $\theta \not\equiv 0$.

We prove Tauberian theorems for random walks with positive drift obeying the central limit theorem. The results include (i) conclusions involving certain averages, relevant to number-theoretic densities and extending results of Diaconis and Stein; (ii) pointwise conclusions, including the classical Borel-Tauber theorem and extending results of Schmaal, Stam and de Vries.

Let $z$ denote a point of $R_2$. We study random flows $Z_{st}(z) \in R_2, 0 \leq s \leq t < \infty, z \in R_2, Z_{tu}(Z_{st}(z)) = Z_{su}(z)$ for $s \leq t \leq u$. Such flows are called Brownian if $Z$ is continuous in $(s, t, z)$ and has appropriate spatial and temporal homogeneity properties and if $Z_{st}, Z_{uv}, \cdots$ are independent homeomorphisms of $R_2$ onto $R_2$ when $s \leq t \leq u \leq v \leq \cdots$. For a Brownian flow the coordinates of any $k$ points are a $2k$-dimensional continuous Markov process, $k = 1, 2, \cdots$. If these processes are diffusions whose diffusion matrices have bounded continuous derivatives of order $\leq 2$ (i.e., are $C^2$-bounded), then the diffusion matrices are necessarily obtained in a certain way from the covariance tensor of the field of infinitesimal displacements. A converse is given in the incompressible isotropic case: given a $C^2$-bounded covariance tensor of an isotropic solenoidal $R_2$-valued field in $R_2$, there exists a corresponding incompressible isotropic Brownian flow.

A class of two-parameter martingales, named "martingales with orthogonal increments" or "martingales of direction independent variation," is introduced. It is shown that this class, which is characterized by a sample function property, is included in the class of martingales of path independent variation and includes the class of strong martingales. The class of martingales with orthogonal increments is stable under stochastic integration and some results, which were obtained previously for strong martingales, hold also for martingales with orthogonal increments. It is shown that if $M_z$ is a martingale with orthogonal increments on the sigma-fields generated by the Wiener process then there exists a Wiener process such that $M_z$ can be represented as a stochastic integral of first type with respect to it.

Let $(E, \mathscr{B}, \mu)$ be a measure space, let $\theta$ be a directed set with a countable cofinal subset, and let $(\mathscr{B}_\tau)_{\tau \in \theta}$ be an increasing family of sub-$\sigma$-algebras of $\mathscr{B}$. A martingale $(f_\tau)_{\tau \in \theta}$ is said to be of semibounded variation whenever the set $\{\int_B f_\tau d\mu \mid \tau \in \theta, B \in \mathscr{B}_\tau\}$ is bounded either from above or below. Denote conditional expectation by $\mathscr{E}$. We show that if every martingale of the form $\mathscr{E}(f \mid \mathscr{B}_\tau)_{\tau\in\theta}$ for some $\mathscr{B}$-measurable function $f$ with $\int|f| d\mu < \infty$ is order convergent, then every martingale of semibounded variation is order convergent. When the family $(\mathscr{B}_\tau)_{\tau\in\theta}$ satisfies a certain refinement condition, we obtain a sufficient condition for order convergence of martingales of semibounded variation in terms of order convergence of martingales which converge stochastically to 0.

Some characterizations of semiorders defined on the set of all probability measures on $R^n$ by the set of Schur-convex functions and by some subsets of all convex functions are proved. A connection of these results to the theorem of Hardy, Littlewood and Polya on the rearrangement of functions is discussed. Furthermore, by means of the results on the ordering of probability measures a generalization of a theorem on doubly stochastic linear operators due to Ryff is proved.

We consider the problem of allocating effort between projects at different stages of development when new projects are also continually appearing. An expression (14) is derived for the expected reward yielded by the Gittins index policy. This is shown to satisfy the dynamic programming equation for the problem, so confirming optimality of the policy.

By a decision process is meant a pair $(X, \Gamma)$, where $X$ is an arbitrary set (the state space), and $\Gamma$ associates to each point $x$ in $X$ an arbitrary nonempty collection of discrete probability measures (actions) on $X$. In a decision process with nonnegative costs depending on the current state, the action taken, and the following state, there is always available a Markov strategy which uniformly (nearly) minimizes the expected total cost. If the costs are strictly positive and depend only on the current state, there is even a stationary strategy with the same property. In a decision process with a fixed goal $g$ in $X$, there is always a stationary strategy which uniformly (nearly) minimizes the expected time to the goal, and, if $X$ is countable, such a stationary strategy exists which also (nearly) maximizes the probability of reaching the goal.

We consider a $K$-dimensional diffusion process $Z$ whose state space is the nonnegative orthant. On the interior of the orthant, $Z$ behaves like a $K$-dimensional Brownian motion with arbitrary covariance matrix and drift vector. At each of the (`K-1) dimensional hyperplanes that form the boundary of the orthant, $Z$ reflects instantaneously in a direction that is constant over that hyperplane. There is no extant theory of multidimensional diffusion that applies to this process, because the boundary of its state space is not smooth. We adopt an approach that requires a restriction on the directions of reflection, but Reiman has shown that this restriction is met by all diffusions $Z$ arising as heavy traffic limits in open $K$-station queuing networks. Our process $Z$ is defined as a path-to-path mapping of $K$-dimensional Brownian motion. From this construction it follows that $Z$ is a continuous Markov process and a semimartingale. Using the latter property, we obtain a change of variable formula from which one can develop a complete analytical theory for the process $Z$.

We consider the stochastic equation $X(t) = W(t) + \beta l^X_0(t)$, where $W$ is a standard Wiener process and $l^X_0(\cdot)$ is the local time at zero of the unknown process $X$. There is a unique solution $X$ (and it is adapted to the fields of $W$) if $|\beta| \leq 1$, but no solutions exist if $|\beta| > 1$. In the former case, setting $\alpha = (\beta + 1)/2$, the unique solution $X$ is distributed as a skew Brownian motion with parameter $\alpha$. This is a diffusion obtained from standard Wiener process by independently altering the signs of the excursions away from zero, each excursion being positive with probability $\alpha$ and negative with probability $1 - \alpha$. Finally, we show that skew Brownian motion is the weak limit (as $n \rightarrow \infty$) of $n^{-1/2}S_{\lbrack nt\rbrack}$, where $S_n$ is a random walk with exceptional behavior at the origin.

Let $Y$ be a nonnegative submartingale. A martingale $M$ is constructed on an enlarged space, with the property that $Y = |M|$.

A family of games $G = G(\sigma, u)$ is defined such that (a) for each $\sigma$ the set of all $u$ for which Player I can force a win in $G(\sigma, u)$ is a Borel set $B(u)$ and (b) every Borel set is a $B(u)$ for some $u$.

Let $\{X_k\}$ be a sequence of independent random variables which are centered at their means; let $\{T_k\}$ be an i.i.d. sequence of $\beta$-dimensional random vectors with common distribution $\mu$; and let $\{X_k\}$ and $\{T_k\}$ be independent. With $\mathscr{L}$ the collection of lower layers, a necessary and sufficient condition for the almost sure convergence of $\sup_{L \in \mathscr{L}}|\sum^n_{k = 1} \chi L (T_k)/n - \mu(L)|$ to zero is given. In addition, this condition on $\mu$ is shown to imply that $\sup_{L \in \mathscr{L}}|\sum^n_{k = 1} X_{k\chi L}(T_k)|/n \rightarrow 0$ a.s. provided the $X_k$ satisfy a first moment-like condition. Rates of convergence are also investigated.

The growth rate at 0 of a Levy process is compared with the growth rate at a local minimum, $m$, of the process. For the lim inf it is found that the growth rate at $m$ is the same as that on the set of "ladder points" following 0, parameterized by inverse local time; this result gives a precise meaning to the notion that a Levy process leaves its minima "faster" than it leaves 0. A less precise result is obtained for the lim sup.

Firstly, a well-known upper estimate concerning the distance of independent products of probability measures is extended to the case of signed measures. The upper bound depends on the total variation of the signed measures and on the distances of the single components where the distances are measured in the sup-metric. Under certain regularity conditions, the upper estimate can be sharpened by using asymptotic expansions. These expansions hold true over the set of all integrable function. An application of these results together with an asymptotic expansion of the distribution of a single order statistic yields an asymptotic expansion of the joint distribution of order statistics under the exponential distribution.

A Berry-Esseen bound of order $n^{-1/2}$ is established for linear combinations of order statistics with smooth weight functions. The underlying distribution function must possess a finite absolute third moment. This improves an earlier result of the author.