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 $F_{i:n}(x)=P{X_{i:n}\leq x}.$ Using majorization theory we write upper and lower bounds for $F$ expressed in terms of mixtures of distribution functions of order statistics, i.e. $\sum \limits_{i=1}^{n}p_{i}F_{i:n}$ and $\sum \limits_{i=1}^{n}p_{i}F_{n-i+1:n}.$ 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 $m\rightarrow\infty.$