Eric Renshaw

Markov branching processes with instantaneous immigration

SummaryMarkov branching processes with instantaneous immigration possess the property that immigration occurs immediately the number of particles reaches zero, i.e. the conditional expectation of sojourn time at zero is zero. In this paper we consider the existence and uniqueness of such a structure. We prove that if the sum of the immigration rates is finite then no such structure can exist, and we provide a necessary and sufficient condition for existence for the case in which this sum is infinite. Study of the uniqueness problem shows that for honest processes the solution is unique.

A CORRELATED RANDOM WALK MODEL FOR TWO-DIMENSIONAL DIFFUSION

A two-dimensional lattice random walk is studied in which, at each stage, the direction of the next step is correlated to that of the previous step. An exact expression is obtained from the characteristic function for the dispersion matrix of the n-step transition probabilities, and the limiting diffusion equation is derived.

A NOTE ON THE RECURRENCE OF A CORRELATED RANDOM WALK

A non-analytical proof of recurrence is obtained via an embedding procedure for a two-dimensional correlated lattice random walk.

The high-order autocovariance structure of the telegraph wave

Exact expressions are developed for the n th order autocovariance structure of the telegraph wave, the integral of which defines a biased correlated random walk.

The spatial distribution of Tribolium confusum

Neyman, Park and Scott (1956) describe an experiment which they performed to determine the spatial distribution of Tribolium confusum developing within a closed container. To explain the concentration of beetles at the boundary a birth-death-migration model is developed in which the beetles may migrate over a set of lattice points, and this is shown to produce a distribution of the required shape. Not only is this distribution independent of the number of lattice points, but it is also indistinguishable from the associated diffusion process. TRIBOLIUM CONFUSUM; SPATIAL POPULATION PROCESS; LATTICE; BIRTH, DEATH AND MIGRATION; DIFFUSION