Stochastic ordering of random kth record values

Let ${\displaystyle X_1,X_2,...}$ be a sequence of independent and identically distributed random variables with continuous distribution function F(x). Denote by X(1,k),X(2,k),... the kth record values corresponding to ${\displaystyle X_1,X_2,...}$ We obtain some stochastic comparison results involving the random kth record values X(N,k), where N is a positive integer-valued random variable which is independent of the ${\displaystyle X_i}$.