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

TL;DR

This paper investigates the asymptotic behavior of the maximum total progenies within a large window in critical Galton-Watson processes, providing new insights into their tail probabilities and expectations.

Contribution

It characterizes the asymptotic behavior of the maximum sum of progenies over a moving window in critical Galton-Watson processes, extending understanding of their tail distributions.

Findings

Asymptotic behavior of EM_m(j) for large m and j/m converging in [0,1]

Tail probability estimates for M_infinity(j)

Behavior of maximum progeny sums in critical branching processes

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).