On distributions of record ranges in nonstandard situations

Abstract

The record theory is an important part of the probability theory and mathematical statistics and has many applications in various fields of knowledge. Most of the known methods deal with the standard model (when the initial random variables are independent and identically distributed). However, various nonstandard models also show some practical interest. In this article, a number of results for distributions and properties of record ranges and sample ranges are obtained for initial sequences of random variables in some nonstandard situations, namely when the variables of the initial sequence have the Laplace distribution or the mixture of two geometric distributions on the positive and negative semi-axes. The converse problem is also investigated, namely: the aforementioned distributions can be characterized by independency of W(1) and W(2) − W(1), where W(1),W(2), . . . are sample ranges in the sequences of initial random variables, if the constant X0 = 0 is added before the initial sequence. Refs 9.