Limit theorems for maxima of independent random sums

We consider extrema of the form
$$ Y_{mn}=\max\limits_{1\le i\le m}\sum^n_{j=1}X_{ij},\quad m,n\ge 1, $$
where $X_{ij}$, $i,j\ge 1$, are independent identically distributed random variables. An asymptotic behavior of $Y_{mn}$ as ${m,n\to\infty}$ is investigated. In particular, the paper shows when the asymptotic behavior of $Y_{mn}$ coincides with the asymptotics of maxima of normally distributed variables under some linear normalizing.