On a maximum sequence in a critical multitype branching process (Q1196936)

The \(p\)-type (positively regular, non-singular) critical branching process, \(\{Z_ n\}\), is considered. Let \(v\) be the right eigenvector of the mean matrix corresponding to the eigenvalue 1 and let \(Y_ n\) be \(Z_ n\cdot v\), which is a martingale. It is the maximum sequence of this martingale that is studied. Specifically if \(M_ n=\max\{Y_ j: 0\leq j\leq n\}\), then, under second moment assumptions, \((\log n)^{-1} E_ i M_ n\to i\cdot v\) where \(E_ i\) is the expectation given that \(i\) is the starting state. (Note that \(i\) is a \(p\)-vector.) The one type result is in [the author, ibid. 16, No. 2, 502-507 (1988; Zbl 0643.60063)] whilst a weaker multitype result is in [\textit{A. Spataru}, J. Appl. Probab. 28, No. 4, 893-897 (1991; Zbl 0745.60087)]. The proof is quite intricate.