Critical Galton-Watson processes: The maximum of total progenies within a large window

2006/01/13 by Klaus Fleischmann, Fleischmann, Klaus, Vladimir Vatutin +4
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #60F17 #60J80 #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:60F17 #msc:60J80

paper · pdf · doi:10.48550/arxiv.math/0601333

28 pages

arxiv created 2006/01/13 · openalex publication_date 2006/01/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider a critical Galton-Watson process Z=Z_n: n=0,1,... 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,...,k+j), and M_m(j) the maximum of the S_k with k moving in [0,m-j]. We describe the asymptotic behavior of the expectation EM_m(j) if the window width j=j_m is such that j/m converges in [0,1] as m tends to infinity. This will be achieved via establishing the asymptotic behavior of the tail probabilities of M_infinity(j).

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