A multiple coupon collection process and its Markov embedding structure

TL;DR

This paper generalizes the coupon collection process by analyzing its Markov embedding structure, providing conditions for representing the process within continuous-time Markov semigroups.

Contribution

It introduces a generalized coupon collection model and derives explicit conditions for its embedding into continuous-time Markov processes.

Findings

Derived explicit conditions for Markov embedding.

Analyzed properties of the generalized coupon collection process.

Provided interpretations of the embedding conditions.

Abstract

The embedding problem of Markov transition matrices into continuous-time Markov semigroups is a classic problem that regained a lot of impetus and activities in recent years. We consider it here for the following generalisation of the well-known coupon collection process: from a finite set of distinct objects, a subset is drawn repeatedly according to some probability distribution, independently and with replacement, and each time united with the set of objects sampled so far. We derive and interpret properties of and explicit conditions for the resulting discrete-time Markov chain to be representable within a semigroup or a flow of a continuous-time process of the same type.