Characterizations of the exponential distribution by some properties of the record values

A sequence {X_n, n≥1} of independent and identically distributed random variables with continuous cumulative distribution function F(x) is considered. X_j is a record value of this sequence if X_j>max {X₁, X₂, ..., X_j?1}. We define L(n)=min {j|j>L(n?1), X_j>X_L(n?1)}, with L(0)=1. Let Z_n,m=X_L(n)?X_L(m), n>m≥0. Some characterizations of the exponential distribution are considered in terms of Z_n,m and X_L(m).