Multiple <i>q</i>-Integral and Records from Geometrically Distributed Sequences

Abstract

We study the distribution theory of some statistics related to records from a sequence of geometrically distributed random variables. The main novelty in this study is the introduction of q-calculus techniques. We obtain representations for the probability functions of some variables of interest, such as, for instance, record times, number of records, inter-record times, record indicators, etc., in terms of multiple q-integrals. It is remarkable to note that the expressions thus obtained are q-analogues of the corresponding well-known results in the absolutely continuous case. We also explore a duality between the results in the case of weak and ordinary records.

Keywords

• weak and ordinary records
• geometric distributions
• Jackson integral
• multiple q-integral
• q-Stirling numbers