Distribution of record statistics in a geometrically increasing population

Abstract

The probability function and binomial moments of the number N n of (upper) records up to time (index) n in a geometrically increasing population are obtained in terms of the signless q-Stirling numbers of the first kind, with q being the inverse of the proportion λ of the geometric progression. Further, a strong law of large numbers and a central limit theorem for the sequence of random variables N n , n = 1 , 2 , … , are deduced. As a corollary the probability function of the time T k of the kth record is also expressed in terms of the signless q-Stirling numbers of the first kind. The mean of T k is obtained as a q-series with terms of alternating sign. Finally, the probability function of the inter-record time W k = T k - T k - 1 is obtained as a sum of a finite number of terms of q-numbers. The mean of W k is expressed by a q-series. As k increases to infinity the distribution of W k converges to a geometric distribution with failure probability q. Additional properties of the q-Stirling numbers of the first kind, which facilitate the present study, are derived.