### On the asymptotic patterns of supercritical branching processes in varying environments

Harry Cohn
Source: Ann. Appl. Probab. Volume 6, Number 3 (1996), 896-902.

#### Abstract

Let ${Z_n}$ be a branching process whose offspring distributions vary with n. It is shown that the sequence ${\max_{i>0} P(Z_n = i)}$ has a limit. Denote this limit by M. It turns out that M is positive only if the offspring variables rapidly approach constants. Let ${c_n}$ be a sequence of constants and $W_n = Z_n / c_n$. It will be proven that $M = 0$ is necessary and sufficient for the limit distribution functions of all convergent ${W_n}$ to be continuous on $(0, \infty)$. If $M > 0$ there is, up to an equivalence, only one sequence ${c_n}$ such that ${W_n}$ has a limit distribution with jump points in $(0, \infty)$. Necessary and sufficient conditions for continuity of limit distributions are derived in terms of the offspring distributions of ${Z_n}$.

First Page:
Primary Subjects: 60J80
Secondary Subjects: 60F25
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.aoap/1034968232
Mathematical Reviews number (MathSciNet): MR1410120
Digital Object Identifier: doi:10.1214/aoap/1034968232
Zentralblatt MATH identifier: 0869.60075

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PARKVILLE, VICTORIA, 3052 AUSTRALIA E-MAIL: h.cohn@stats.unimelb.edu.au