For modelling non-stationary spatial random fields Z = {Z(x) : x∊ℝn, n≥2} a recent method has been proposed to deform bijectively the index space so that the spatial dispersion D(x,y) = var[Z(x)-Z(y)], (x,y)∊ℝnxℝn, depends only on the Euclidean distance in the deformed space through an isotropic variogram γ. We prove uniqueness of this model in two different cases: (i) γ is strictly increasing; (ii) γ(u) is differentiable for u > 0.

We study the expected time for the work in an M/G/1 system to exceed the level x, given that it started out initially empty, and show that it can be expressed solely in terms of the Poisson arrival rate, the service time distribution and the stationary delay distribution of the M/G/1 system. We use this result to construct an efficient simulation procedure.

We provide a probabilistic proof of the Stein's factors based on properties of birth and death Markov chains, solving a tantalizing puzzle in using Markov chain knowledge to view the celebrated Stein–Chen method for Poisson approximations. This work complements the work of Barbour (1988) for the case of Poisson random variable approximation.

We prove a sharp upper bound on the number of patterns that can be stored in the Hopfield model if the stored patterns are required to be fixed points of the gradient dynamics. We also show corresponding bounds on the one-step convergence of the sequential gradient dynamics. The bounds coincide with the known lower bounds and confirm the heuristic expectations. The proof is based on a crucial idea of Loukianova (1997) using the negative association properties of some random variables arising in the analysis.

We consider a classical population flow model in which individuals pass through n strata with certain state-dependent probabilities and at every time t = 0,1,2,…, there is a stochastic inflow of new individuals to every stratum. For a stationary inflow process we prove the convergence of the joint distribution of group sizes and derive the limiting Laplace transform.