Characterizations of Pareto distribution by the assumption of identical distributions on upper record values

Let {X-k, k >= 1} be a sequence of i.i.d. random variables which has absolutely continuous distribution function F such that F (1) = 0 and F (x) < 1 for all x > 1. We show that if F is an element of C-1(,) alternatively, F is an element of C-2 or F is an element of C-3(,) then X-k's have the Pareto distribution if and only if Wn+1, (n) has an identical distribution with Xk for all n = 1, alternatively, W-n+1,W-n has an identical distribution with W-n,W- n- 1 for all n >= 2 or X-U ((n+ 1)) and X-U ((n)) . U are identically distributed, U is independent of X-U(n) and XU(n+ 1), and U is distributed as X-k's for all n >= 1.