Critical Galton–Watson process: The maximum of total progenies within a large window

Consider a critical Galton–Watson process $Z=\{Z_n:n=0,1,\dots\}$ of index $1+\alpha$, $\alpha\in(0,1]$. Let $S_k(j)$ denote the sum of the $Z_{n}$ with $n$ in the window $[k,\dots,k+j)$ and let $M_{m}(j)$ be the maximum of the $S_{k}(j)$ with $k$ moving in $[0,m-j]$. We describe the asymptotic behavior of the expectation $\mathbf{E}M_m(j)$ if the window width $j=j_{m}$ is such that $j/m\to\eta\in$ $[0,1]$ as $m\uparrow\infty$. This will be achieved via establishing the asymptotic behavior of the tail of the distribution of the random variable $M_{\infty}(j)$.