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Consider a critical Galton-Watson process Z={Z_n: n=0,1,...} of index1+alpha, alpha in (0,1]. Let S_k(j) denote the sum of the Z_n with n in thewindow [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 windowwidth j=j_m is such that j/m converges in [0,1] as m tends to infinity. Thiswill be achieved via establishing the asymptotic behavior of the tailprobabilities of M_{infinity}(j).

Critical Galton–Watson Processes: The Maximum of Total Progenies within a Large Window