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On the maximum sequence in a critical branching process. (English) Zbl 0643.60063

Let \(\{Z_ n\}\) be a critical branching process in which the family size has finite variance. Let \(M_ n\) be the maximum population size up to the n th generation, so \(M_ n=\max \{Z_ j:0\leq j\leq n\}\). It is a consequence of results of A. G. Pakes [J. Appl. Probab. 24, 768-772 (1987; Zbl 0636.60047)] that, if \(Z_ 0=i\), lim sup(E(M\({}_ n)/\log n)\leq i\). Here it is shown that, if \(Z_ 0=i\), \(E(M_ n)/\log n\to i\) as \(n\to \infty\).
Reviewer: J.Biggins

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K99 Special processes

Citations:

Zbl 0636.60047
Full Text: DOI