On the Distribution and Moments of Record Values in Increasing Populations

Consider a sequence of n independent observations froma population of increasing size qi, i = 1,2, ... and an absolutelycontinuous initial distribution function. the distribution of the kthrecord value is represented as a countable mixture, with mixing thedistribution of the kth record time and mixed the distribution of thenth order statistic. precisely, the distribution function and (power)moments of the kth record value are expressed by series, with coefficientsbeing the signless. generalized stirling numbers of the firstkind. then, the probability density function and moments of the kthrecord value in a geometrically increasing population are expressed byq-series, with coefficients being the signless q-stirling numbers of thefirst kind. in the case of a uniform distribution for the initial population,two equivalent q-series expressions of the jth (power) momentof the kth record value are derived. also, the distribution functionand the moments of the kth record value in a factorially increasingpopulation are deduced.