On the total time spent in records by a discrete uniform sequence (Q2774455)

The authors consider the sum \(S_d\) of record values in a sequence of random variables that are uniformly distributed on \(1,2,\dots,d\). \(S_d\) can be interpreted as the total amount of time spent in record lifetimes in the standard renewal theory. The aim of the paper is to investigate the distributional limit of \(S_d\) and some related quantities, as \(d\to\infty\). Instead of a (functional) analytic methodology and recursion relations for the probabilities of interest, the authors provide a more probabilistic approach, using an almost sure construction based on a suitable background stochastic process that drives a whole family of models. Some explicit values are given for the case \(d=6\), that can be interpreted as a simple game of chance (the standard model for the repeated toss of a fair die).