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This article is cited in 10 scientific papers (total in 10 papers)
Critical Galton–Watson process: The maximum of total progenies within a large window
V. A. Vatutina, V. I. Vakhtel'b, K. Fleischmannc a Steklov Mathematical Institute, Russian Academy of Sciences
b Technische Universität München
c Weierstrass Institute for Applied Analysis and Stochastics
Abstract:
Consider a critical Galton–Watson process $Z=\{Z_n:n=0,1,\dots\}$ of index $1+\alpha$, $\alpha\in(0,1]$. Let $S_k(j)$ denote the sum of the $Z_{n}$ with $n$ in the window $[k,\dots,k+j)$ and let $M_{m}(j)$ be the maximum of the $S_{k}(j)$ with $k$ moving in $[0,m-j]$. We describe the asymptotic behavior of the expectation $\mathbf{E}M_m(j)$ if the window width $j=j_{m}$ is such that $j/m\to\eta\in$ $[0,1]$ as $m\uparrow\infty$. This will be achieved via establishing the asymptotic behavior of the tail of the distribution of the random variable $M_{\infty}(j)$.
Keywords:
branching of index one plus alpha, limit theorem, conditional invariance principle, tail asymptotics, moving window, maximal total progeny, lower deviation probabilities.
Received: 16.01.2006 Revised: 02.04.2007
Citation:
V. A. Vatutin, V. I. Vakhtel', K. Fleischmann, “Critical Galton–Watson process: The maximum of total progenies within a large window”, Teor. Veroyatnost. i Primenen., 52:3 (2007), 419–445; Theory Probab. Appl., 52:3 (2008), 470–492
Linking options:
https://www.mathnet.ru/eng/tvp72https://doi.org/10.4213/tvp72 https://www.mathnet.ru/eng/tvp/v52/i3/p419
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