Geodesic
The sum and the order statistics of independent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 4, pp. 710-727

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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 $\{\mathbf P(X_n$ and the sequences of distribution functions $\displaystyle\{\mathbf P(\min_{1\le j\le k_n}X_{nj}$ and $\displaystyle\{\mathbf P(\max_{1\le j\le k_n}X_{nj}$. The limit theorems in Lévy's metrics, the conditions for the convergence of moments and the global limit theorems are proved. 
@article{TVP_1979_24_4_a2,
     author = {V. M. Kruglov},
     title = {The sum and the order statistics of independent random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {710--727},
     publisher = {mathdoc},
     volume = {24},
     number = {4},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1979_24_4_a2/}
}
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V. M. Kruglov. The sum and the order statistics of independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 4, pp. 710-727. http://geodesic.mathdoc.fr/item/TVP_1979_24_4_a2/