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).Comment: 28 page