Characterizations of the pareto distribution by conditionalexpectations of record values

Abstract

Let $ X_1 , X_2 , \cdots $ be a sequence of independent and identically distributed random variables with continuous cumulative distribution function $ F(x).$ $ X_j $ is an upper record value of this sequence if $ X_j > \max\{ X_1 , X_2 , \cdots , X_{j-1} \}.$ We define $ u(n) = \min \{j \vert j$ $ > u(n-1), X_j > X_{u(n-1)}, n \geq 2 \} $ with $ u(1) = 1.$ Then $ F(x)$ $=1-x^{\theta},\, \, x>1, \,\, \theta<-1$ if and only if $(\theta+1)E[X_{u(n+1)}|X_{u(m)}$ $=y] = \theta E[X_{u(n)}| X_{u(m)}=y],$ $(\theta+1)^2 E[X_{u(n+2)}|X_{u(m)}=y] = \theta^2 E[X_{u(n)}| X_{u(m)}=y]\,, $ or $ (\theta+1)^3 E[X_{u(n+3)}|X_{u(m)}=y] = \theta^3 E[X_{u(n)}| X_{u(m)}=y],$\,\, $n \ge m+1 .$