Limit theorems for maxima of some dependent random sums

A family of extrema having the form $$Y_{mn} = mathop {max }limits_{1 leqslant i leqslant m} sumlimits_{j = 1}^n {X_{ij} , m,n geqslant 1,}$$ is considered, here the random variables {X _ij }, i ?? 1, j ?? 1, are dependent in columns (with identical j) and independent in rows (with different j). The asymptotics of Y _mn for m, n ?? ?? is studied. Three particular cases are considered: a normal distribution, a Laplace distribution, and an ??-stable distribution.