We propose a model of a passive nerve cylinder undergoing random stimulus along its length. It is shown that this model is approximated by the solution of a stochastic partial differential equation. Numerous properties of the sample paths are derived, such as their modulus of continuity, quadratic and quartic variation, and it is shown that the solution exhibits the phenomenon of flicker noise. The first-passage problem is studied, and it is shown to be connected with a first-hitting time for an infinite-dimensional diffusion.

The wind exerts a sorting effect on particles of sands and, under certain stable conditions that occur frequently but whose nature is little understood, the sorting results in log-size distributions of the hyperbolic form, first noted by R. A. Bagnold. Here, for wind-tunnel experiments a stochastic model is constructed which exhibits a sorting effect deriving from the dependence of distance travelled on the size of the single particle. Under rather specific, experimentally testable assumptions the model reproduces log-size distributions which are of the hyperbolic type and show a variation with distance along the wind tunnel that accords with experimental findings of R. A. Bagnold.

The two common critical probabilities for a lattice graph L are the cluster size critical probability pH(L) and the mean cluster size critical probability pT(L). The values for the honeycomb lattice H and the triangular lattice T are proved to be pH(H) = pT(H) = 1–2 sin (π/18) and PH(T) = pT(T) = 2 sin (π/18). The proof uses the duality relationship and the star-triangle relationship between the two lattices, to find lower bounds for sponge-crossing probabilities.

In the bond percolation process on the square lattice, with let S(k) be the probability that some open path joins the longer sides of a sponge with dimensions k by a log k. There exists a positive constant α = αp such that Consequently, the subset of the square lattice {(x, y):0 ≦ y ≦ f(x)} which lies between the curve y = f(x) and the x-axis has the same critical probability as the square lattice itself if and only if f(x)/log x → ∞ as x → ∞.

In this paper we continue our investigation of entropy comparisons with emphasis on multivariate distributions. For multiparameter cases of the multinomial and negative multinomial distributions we consider various higher-order forms of multivariate convexity. For the multinormal, Wishart, and t-distributions we define a partial ordering on the set of covariance matrices and determine monotonicity of the entropy functional. We further indicate some entropy inequalities for different sampling schemes. Because of the complex nature of multivariate partial ordering relations several problems remain open.

Let X be a Markov process on the line. Under certain conditions it is possible to find a diffusion process which is an approximation to X in the following sense:

(1) X can be embedded in that is there are stopping times (Tt) such that {Xt, t ≥ 0} and have the same distribution;

(2) for each t, E{Tt} = t.

We call the well-timed diffusion approximation to X, and suggest that it is useful for approximating quantities like the first-exit probabilities and expected first-exit times of X.

We determine the well-timed approximation in several special cases and give an asymptotic approximation for use in cases in which cannot be exactly determined.

The general framework of a Markov chain in a random environment is presented and the problem of determining extinction probabilities is discussed. An efficient method for determining absorption probabilities and criteria for certain absorption are presented in the case that the environmental process is a two-state Markov chain. These results are then applied to birth and death, queueing and branching chains in random environments.

In [5] Doeblin considered some classes of finite non-homogeneous Markov chains and gave without proofs several results concerning their asymptotic behaviour. In the present paper we first attempt to make Doeblin's results precise and try to reconstruct his arguments. Subsequently we investigate more general situations, where a state space decomposition is provided by the sets occurring in the representation of the atomic sets of the tail σ-field. We show that Doeblin's notion of an associated chain, as well as considerations regarding the tail σ-field structure of the chain, can be used to solve such cases.

We derive conditions for the time-uniform continuity of random sequences with respect to variations of governing parameters, and also obtain some estimates of the modulus of continuity. We then apply the results to find conditions of continuity for Markov chains with arbitrary state space, and to construct finite approximations to them.

We extend the optimal strategy results of Kelly and Breiman and extend the class of random variables to which they apply from discrete to arbitrary random variables with expectations. Let Fn be the fortune obtained at the nth time period by using any given strategy and let Fn∗ be the fortune obtained by using the Kelly–Breiman strategy. We show (Theorem 1(i)) that Fn/Fn∗ is a supermartingale with E(Fn/Fn∗) ≤ 1 and, consequently, E(lim Fn/Fn∗) ≤ 1. This establishes one sense in which the Kelly–Breiman strategy is optimal. However, this criterion for ‘optimality’ is blunted by our result (Theorem 1(ii)) that E(Fn/Fn∗) = 1 for many strategies differing from the Kelly–Breiman strategy. This ambiguity is resolved, to some extent, by our result (Theorem 2) that Fn∗/Fn is a submartingale with E(Fn∗/Fn) ≤ 1 and E(lim Fn∗/Fn) ≤ 1; and E(Fn∗/Fn) = 1 if and only if at each time period j, 1 ≤ j ≤ n, the strategies leading to Fn and Fn∗ are ‘the same’.