On independence of $k$-record processes: Ignatov's theorem revisited

Abstract

For an infinite sequence of independent and identically distributed (i.i.d.) random variables, the k-record process consists of those terms that are the kth largest at their appearance. Ignatov's theorem states that the k-record processes, $k = 1, 2, \dots ,$ are i.i.d. A new proof is given which is based on a "continualization" argument. An advantage of this fairly simple approach is that Ignatov's theorem can be stated in a more general form by allowing for different tiebreaking rules. In particular, three tiebreakers are considered and shown to be related to Bernoulli, geometric and Poisson distributions.