Random Translation of Records from Geometric Distributions

It is well known that the distributions of two records from a sequence of iid geometric random variables are related by a random translation. The distribution of the random translation is a negative binomial distribution. In this article, we investigate if this property characterizes the geometric parent distribution. We show that the answer is positive when weak records are considered. On the other hand, when ordinary records are considered, it is possible to characterize tail-geometric distributions if the records are consecutive, but in the case of non consecutive ordinary records it is possible to find out distributions that do not seem to be tail-geometric satisfying the property of random translation.