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

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Consider a critical Galton-Watson process Z=Zₙ: n=0,1,... of index 1+alpha, alpha in (0,1]. Let Sₖ(j) denote the sum of the Zₙ with n in the window [k,...,k+j), and Mₘ(j) the maximum of the Sₖ with k moving in [0,m-j]. We describe the asymptotic behavior of the expectation EMₘ(j) if the window width j=jₘ 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 Minfinity(j).