On the maximum sequence in a critical branching process
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 \textit{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 \(n\to \infty\).
scientific article; zbMATH DE number 4048800