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A multiple coupon collection process and its Markov embedding structure

Ellen Baake, Michael Baake

TL;DR

The paper addresses the problem of embedding the discrete-time MCCP transition matrix into a continuous-time Markov semigroup. It reveals a rich algebraic structure built from CM and CG matrices within the incidence algebra, and uses Möbius inversion and spectral theory to derive necessary and sufficient embeddability conditions. The principal matrix logarithm R, with CG properties, is the key object: M is embeddable if and only if R is a Markov generator, i.e., all CG parameters r_K are nonnegative (for nonempty K). When the spectrum is simple, R is unique, giving a complete embedding criterion; otherwise, embeddings are constrained by centraliser considerations. The work connects probabilistic MCCP behavior to a precise geometric and algebraic framework, yielding explicit forward-flow representations M(t) = e^{tQ} and a clear probabilistic interpretation of the embedding constraints.

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.

A multiple coupon collection process and its Markov embedding structure

TL;DR

The paper addresses the problem of embedding the discrete-time MCCP transition matrix into a continuous-time Markov semigroup. It reveals a rich algebraic structure built from CM and CG matrices within the incidence algebra, and uses Möbius inversion and spectral theory to derive necessary and sufficient embeddability conditions. The principal matrix logarithm R, with CG properties, is the key object: M is embeddable if and only if R is a Markov generator, i.e., all CG parameters r_K are nonnegative (for nonempty K). When the spectrum is simple, R is unique, giving a complete embedding criterion; otherwise, embeddings are constrained by centraliser considerations. The work connects probabilistic MCCP behavior to a precise geometric and algebraic framework, yielding explicit forward-flow representations M(t) = e^{tQ} and a clear probabilistic interpretation of the embedding constraints.

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.
Paper Structure (6 sections, 15 theorems, 72 equations)

This paper contains 6 sections, 15 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The multiple coupon collection process
  4. Markov matrices and generators for MCCPs
  5. Intermezzo: Algebraic and analytic properties of the model
  6. Positivity conditions and embeddability

Key Result

Lemma 2.2

Let $S$ be a non-empty finite set and let $f : 2^{S} \setminus \varnothing \longrightarrow \mathbb{R}\space_{+}$ be a strictly positive function. Assume that $f$ is extended to a function on the partitions of $S$ by setting $f(\mathcal{A}) = \prod_{A\in\mathcal{A}} f(A)$ for any $\mathcal{A}\in \mat

Theorems & Definitions (35)

  • proof
  • Lemma 2.2
  • proof
  • Definition 3.1
  • proof
  • Example 3.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 25 more