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Volume 8 - June 1976

Contents

Page 3 of 8

Symposium on mathematical genetics, Liverpool, 5–6 April 1976

I. Invited Review and Research Papers

Fifth Conference on Stochastic Processes and their Applications

Correction

Symposium on Mathematical Genetics Edinburgh, 7–8 April 1975

Front Matter

I. Invited Review and Research Papers

Fifth Conference on Stochastic Processes and their Applications

Symposium on Mathematical Genetics Edinburgh, 7–8 April 1975

Symposium on mathematical genetics, Liverpool, 5–6 April 1976

Back Matter

Symposium on mathematical genetics, Liverpool, 5–6 April 1976

I. Invited Review and Research Papers

Fifth Conference on Stochastic Processes and their Applications

Symposium on Mathematical Genetics Edinburgh, 7–8 April 1975

I. Invited Review and Research Papers

Fifth Conference on Stochastic Processes and their Applications

Conference on Patterns and Processes in the Plane: Swansea, 25–27 September 1975

Research Article

  • Stochastic models of grouping changes

  • Byron J. T. Morgan
  • Published online by Cambridge University Press:
    01 July 2016, pp. 30-57

  • Several simple stochastic models are given for a finite closed system of individuals existing in clusters which may come together to form larger clusters which may in turn split up. Some of these models are analysed and compared in equilibrium. Several of the models fit into a general framework established and investigated by Whittle (1965a); it is shown that these models have an equilibrium solution of a particularly simple form, deduced by Whittle, if and only if the models are stochastically reversible. A normal approximation to two of the models in equilibrium is found to give the same mean value as a deterministic approximation.

  • Recurrence criteria for multi-dimensional Markov chains and multi-dimensional linear birth and death processes

  • Harry Kesten
  • Published online by Cambridge University Press:
    01 July 2016, pp. 58-87

  • Criteria are established for a discrete-time Markov process {Xn}n≧0 in Rd to have strictly positive, respectively zero, probability of escaping to infinity. These criteria are mainly in terms of the mean displacement vectors μ(y) = E{Xn+1|Xn = y} – y, and are essentially such that they force a deterministic process w.p.1 to move off to infinity, respectively to return to a compact set infinitely often. As an application we determine of most two-dimensional birth and death processes with rates linearly dependent on the population, whether they can escape to infinity or not.

I. Invited Review and Research Papers

Fifth Conference on Stochastic Processes and their Applications

Conference on Patterns and Processes in the Plane: Swansea, 25–27 September 1975

I. Invited Review and Research Papers

Fifth Conference on Stochastic Processes and their Applications

Research Article

  • A generalization of Goldstein's comparison lemma and the exponential limit law in critical Crump-Mode-Jagers branching processes

  • John M. Holte
  • Published online by Cambridge University Press:
    01 July 2016, pp. 88-104

  • Let Z(t) be the population size at time t in a general age-dependent branching process (as defined by Crump and Mode, or Jagers) in which the number N of offspring of a parent has expected value 1 (critical case). Assuming positivity and finiteness of the second moments of N, of the lifespan distribution and of the expected number of births per parent as a function of age (also assumed to be strongly non-lattice), the distribution of Z(t)/t conditioned on non-extinction at time t is asymptotically exponential. The main step in the proof is a comparison lemma for the probability generating functions of Z(t) and of the embedded generation process.

Page 3 of 8