POPULATION-SIZE-DEPENDENT BRANCHING PROCESS WITH LINEAR RATE OF GROWTH.

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Abstract

The process we consider is a binary splitting, where the probability of division, p//i equals one-half plus 1/2i, i equals 1,2, . . . , depends on the population size, 2i. We show that Z//n converges to infinity almost surely on set Q OVER BAR of positive probability. Z//n/N converges in distribution to a proper limit, SIGMA ** infinity //n// equals //0(1/Z//n) diverges almost surely on Q OVER BAR , SIGMA ** infinity //n// equals //0(1/Z**2//n) converges almost surely on Q OVER BAR and there are no constants C//n such that Z//n/c//n converges in probability to a non-degenerate limit.

Original languageEnglish
Pages (from-to)242-250
Number of pages9
JournalJournal of Applied Probability
Volume20
Issue number2
DOIs
Publication statusPublished - 1 Jan 1983

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