On the Distribution and Moments of Record Values in Increasing Populations

Consider a sequence of n independent observations from a population of increasing size Qi, i = 1,2, ... and an absolutely continuous initial distribution function. The distribution of the kth record value is represented as a countable mixture, with mixing the distribution of the kth record time and mixed the distribution of the nth order statistic. Precisely, the distribution function and (power) moments of the kth record value are expressed by series, with coefficients being the signless. generalized Stirling numbers of the first kind. Then, the probability density function and moments of the kth record value in a geometrically increasing population are expressed by q-series, with coefficients being the signless q-Stirling numbers of the first kind. In the case of a uniform distribution for the initial population, two equivalent q-series expressions of the jth (power) moment of the kth record value are derived. Also, the distribution function and the moments of the kth record value in a factorially increasing population are deduced.