Majorization bounds for distribution function

Abstract

Let $X$ be a random variable with distribution function $F,$ and $X_1,X_2,...,X_n$ are independent copies of $X.$ Consider the order statistics $X_{i:n},$ $i=1,2,...,n$ and denote $F_{i:n}(x)=P\{X_{i:n}\le x\}.$ Using majorization theory we write upper and lower bounds for $F$ expressed in terms of mixtures of distribution functions of order statistics, i.e. $\sum _{i=1}^n{p}_i{F}_{i:n}$ and $\sum _{i=1}^n{p}_i{F}_{n-i+1:n}.$ It is shown that these bounds converge to $F$ \ for a particular sequence $(p_1(m),p_2(m),...,p_n(m)),m=1,2,..$ as $m\to \infty .$

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