Distribution of record statistics in a geometrically increasing population

The probability function and binomial moments of the number _Nn$N_n$ 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$\lambda$ of the geometric progression. Further, a strong law of large numbers and a central limit theorem for the sequence of random variables _Nn$N_n$, n=1,2,…,$n=1,2,\dots ,$ are deduced. As a corollary the probability function of the time _Tk$T_k$ of the kth record is also expressed in terms of the signless q  -Stirling numbers of the first kind. The mean of _Tk$T_k$ is obtained as a q  -series with terms of alternating sign. Finally, the probability function of the inter-record time _Wk=_Tk-_Tk-1$W_k=T_k-T_{k-1}$ is obtained as a sum of a finite number of terms of q  -numbers. The mean of _Wk$W_k$ is expressed by a q-series. As k   increases to infinity the distribution of _Wk$W_k$ 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.