The sum and the order statistics of independent random variables

Let $\{X_n\}$ be a sequence of sums of independent random variables:
$$ X_n=X_{n1}+X_{n2}+\dots+X_{nk_n},\qquad n=1,2,\dots $$
We investigate the connections between the sequence of distribution functions $\{\mathbf P(X_n<u)\}$ and the sequences of distribution functions $\displaystyle\{\mathbf P(\min_{1\le j\le k_n}X_{nj}<u)\}$ and $\displaystyle\{\mathbf P(\max_{1\le j\le k_n}X_{nj}<u)\}$. The limit theorems in Lévy's metrics, the conditions for the convergence of moments and the global limit theorems are proved.