Estimates for the expectation of the maximum of a critical Galton-Watson process on a finite interval (Q2773617)

Let \(Z_n\) be a critical Galton-Watson branching process with \(Z_0=1\). In particular, \(\mathbf{E}Z_1=1\). The authors study the asymptotical behavior of the expectation \(\mathbf{E}M_n\), where \(M_n=\max_{0\leq k\leq n}Z_k\). The main results are stated as follows:NEWLINENEWLINENEWLINE(i) if \(\mathbf{E}Z_1\log^+Z_1\) is finite, then \(\liminf_{n\to\infty}\mathbf{E}M_n\log^{-1}n\geq 1/2\);NEWLINENEWLINENEWLINE(ii) if \(\mathbf{E}Z_1(\log^+Z_1)^\beta\) is finite for some \(\beta>1\), then \(\liminf_{n\to\infty}\mathbf{E}M_n\log^{-1}n\geq \beta/(\beta+1)\);NEWLINENEWLINENEWLINE(iii) if \(\mathbf{E}Z_1(\log^+Z_1)^\beta\) is finite for some \(\beta>2\), then \(\limsup_{n\to\infty}\mathbf{E}M_n\log^{-1}n\leq\beta/(\beta-2)\).NEWLINENEWLINENEWLINEAssertions (ii) and (iii) imply the following corollary: if \(\mathbf{E}Z_1(\log^+Z_1)^\beta<\infty\) for any \(\beta>0\), then \(\mathbf{E}M_n\sim \log n\) as \(n\to\infty\). This corollary generalizes the corresponding result by \textit{V. A. Vatutin} and \textit{V. A. Topchij} [Theory Probab. Appl. 42, No. 1, 17-27 (1997); translation from Teor. Veroyatn. Primen. 42, No. 1, 21-34 (1997; Zbl 0909.60076)].