Concomitants and majorization bounds for bivariate distribution function

Let ($X,Y)$ be a random vector with distribution function $F(x,y),$ and $(X_1,Y_1),(X_2,Y_2),\mathrm{.}\mathrm{.}\mathrm{.},(X_n,Y_n)$ are independent copies of ($X,Y)\mathrm{.}$ Let $X_{i:n}$ be the $i$th order statistics constructed from the sample $X_1,X_2,\mathrm{.}\mathrm{.}\mathrm{.},X_n$ of the first coordinate of the bivariate sample and $Y_{[i:n]}$ be the concomitant of $X_{i:n}\mathrm{.}$ Denote $F_{i:n}% (x,y)=P{X_{i:n}\leq x,Y_{[i:n]}\leq y}.$ Using majorization theory we write upper and lower bounds for $F$ expressed in terms of mixtures of joint distributions of order statistics and their concomitants, i.e. ${\dsum \limits_{i=1}^{n}}% {\sum\limits_{i=1}^{n}} p_{i}F_{i:n}(x,y)$and${\dsum \limits_{i=1}^{n}}% {\sum\limits_{i=1}^{n}} p_{i}F_{n-i+1:n}(x,y).$ It is shown that these bounds converge to $F$ for a particular sequence $(p_1(m),p_2(m),\mathrm{.}\mathrm{.}\mathrm{.},p_n(m)),m=1,2,\mathrm{.}\mathrm{.}$ as $m\to \mathrm{\infty }\mathrm{.}$