Some characterizations of geometric tail distributions based on record values

Abstract

Let Rj be the jth upper record value from an infinite sequence of independent identically distributed positive integer valued random variables. We show that their common distribution must have geometric tail if Rj+k−Rj and Rj are partially independent for some j≥1 and k≥1 or if E(Rj+2−Rj+1| Rj) is a constant. Three versions of partial independence, each of which provides a characterization of the geometric tail are presented.