Limit theorems for the critical age-dependent branching process with infinite variance |
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Authors: | Martin I. Goldstein Fred M. Hoppe |
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Affiliation: | Départment de Mathématiques, Université de Montréal, Montréal, Québec H3C 3J1, Canada;Department of Mathematics, University of Alberta, Edmonton T6G2G1, Alberta, Canada |
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Abstract: | Let Z(t) be the population at time t of a critical age-dependent branching process. Suppose that the offspring distribution has a generating function of the form f(s) = s + (1 ? s)1+αL(1 ? s) where α ∈ (0, 1) and L(x) varies slowly as x → 0+. Then we find, as t → ∞, (P{Z(t)> 0})αL(P{Z(t)>0})~ μ/αt where μ is the mean lifetime of each particle. Furthermore, if we condition the process on non-extinction at time t, the random variable P{Z(t)>0}Z(t) converges in law to a random variable with Laplace-Stieltjes transform 1 - u(1 + uα)?1/α for u ?/ 0. Moment conditions on the lifetime distribution required for the above results are discussed. |
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Keywords: | age-dependent branching process critical branching process extinction probability exponential limit law |
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