Limit theorems for maxima of some dependent random sums

A family of extrema having form
$$ Y_{mn}=\max_{1\le i \le m}\sum_{j=1}^n X_{ij},\qquad m,n\ge1, $$
is considered, here the random variables $\{X_{ij}\}$, $i\ge1$, $j\ge1$, are dependent by columns (with identical $j$) and independent by rows (with different $j$). The asymptotics of $Y_{mn}$ for $m,n\to\infty$ is studied. Three particular cases are considered: a normal distribution, a Laplace distribution, and an $\alpha$-stable distribution.