LIMITING BEHAVIOR FOR AGE AND POSITIONDEPENDENT BRANCHING PROCESSES.
Abstract
In this paper a model is studied for the population transition probabilities for a branching process composed of particles diffusing in a finite interval. The model is in general non-Markovian assuming the branching transformation probabilities for a particle depend on its age and position. The process is described by the random number N sub t (x) of particles in the interval I at time t that are generated by a single particle initially at the point x in I. By considering N sub t (x) as a regenerative process with respect to the random age and position of the initial particle when it is transformed, a functional equation is developed for the generating function for N sub t (x). This functional equation is the basis for the study of the population probabilities P(N sub t (x) = n), n = 0,1,2, ... , as function of x in I and t in (0, infinity). The principal results develop the behavior of N sub t (x) for large t and the dependence of such behavior on x. A convergence is established in distribution for those processes for which N sub t (x) can increase without bound with positive probability as t approaches infinity. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Jun 01, 1964
- Accession Number
- AD0605638
Entities
People
- Howard E. Conner
Organizations
- University of Wisconsin–Madison