Let ξ_1,...,ξ_n be independent uniformly distributed random variables with a distribution function F(x). Let ξ_1~((n)) ≤ ξ_2~((n)) ≤ ··· ≤ ξ_(n-1)~((n)) ≤ ξ_n~((n)) be variational series, constructed by random variables ξ_1,...,£_n. In this article, limit distributions of the kth term(kth order statistics) of variational series ξ_k~((n)) , where n → ∞ and k dependent on the size of a sample n: k = k(n), are considered. It is accepted to call a relation k(n) the rank of the order statistics ξ_(k(n))~((n)). If k(n) → λ,n→ ∞, and 0 < λ < 1, then we speak about central order statistics ξ_k~((n)). If k(n) = const or n - k(n) = const, then appropriate order statistics are called extreme. The asymptotic behavior of external and central order statistics is different. The case k(n) → ∞, k(n) → 0 (or n - k(n) → ∞, k(n) → 1), n → ∞, takes an intermediate place, and in such a situation we speak about intermediate ranks of appropriate order statistics. The asymptotic behavior of extreme order statistics is exhaustively described in and. A description of the asymptotic behavior of central order statistics is contained in. The asymptotic behavior of intermediate order statistics with numbers k(n), satisfying the relations.
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