Limit Probabilities in a Multi-Type Critical Age-Dependent Branching Process
Abstract
Let Z sub i (t) = (Z sub i1 (t), Z sub i2 (t),...,Z sub im(t)) where Z sub ij (t) = number of cells of type j at t starting at t = 0 with one new cell of type i, with 1 < or = i, j < or = m. Assuming this to be an m-type critical age-dependent branching process, for k = (k sub 1,...,k sub m) an m- vector of non-negative integers, not all of which are zero, it is shown that as t approaches infinity, P(Z sub i (t) = k) approx. = c/t squared for some c > 0. Similarly, let N sub i (t) = sub i1 (N sub i1 (t)), N sub i2 (t),...,N sub im(t) ) denote the m-vector with entries N sub ij (t) = total progeny of type j born by t in the critical m-type process starting with a new cell of type i at t = 0. For k = (k sub 1,...,k sub m) an m-vector of non-negative intergers, it is shown that as t approaches infinity, P(N sub i1 (t) = k) approx. = C > 0 where the constants may be evaluated explicitly by a recursion. Related results on marginal probabilities are indicated.
Document Details
- Document Type
- Technical Report
- Publication Date
- Mar 07, 1977
- Accession Number
- ADA044898
Entities
People
- Howard J. Weiner
Organizations
- Stanford University