On equal-input and monotone Markov matrices

Abstract

The practically important classes of equal-input and of monotone Markov matrices are revisited, with special focus on embeddability, infinite divisibility, and mutual relations. Several uniqueness results for the classic Markov embedding problem are obtained in the process. To achieve our results, we need to employ various algebraic and geometric tools, including commutativity, permutation invariance, and convexity. Of particular relevance in several demarcation results are Markov matrices that are idempotents.

Notes on Markov embedding

“…Markov matrices have a special spectral structure as follows, which we recall from [4]. Throughout, we use M d to denote the subset of Markov matrices in Mat(d, R), which is a closed convex set, and A (0) d for the set of all matrices from Mat(d, R) with zero row sums.…”

“…Both types of embeddability require M to be non-singular, via det(e Q ) = e tr(Q) and Eq. ( 10) in the first case and an application of Liouville's theorem to the Kolmogorov forward equation, Ṁ (t) = M (t)Q(t), in the latter; compare [5,Rem. 3.3].…”

“…K=J p K (t) for each t 0 as in (3), and thus a simple, explicit approach to the semigroup. ♦ When, in Theorem 4.7, we hit a situation where more than one generator exists for M (which can only happen for the non-generic case of multiple eigenvalues, and even then at most for very small values of det(M ); compare [8,4] and references therein), only one can have Property PG. In this case, two (or more) Markov semigroups cross each other in M , but the extra solutions rarely have an interesting probabilistic interpretation.…”

“…Our interest here is in the discrete-time Markov chain (X n ) n∈N 0 on the set of subsets of S, where X n is the set of types collected until step n. While we do not consider the usual stopping time problem, we are concerned with whether X is embeddable, that is, whether there is a continuous-time Markov chain (Y t ) t∈R 0 such that P(Y 1 = J | Y 0 = I) = P(X 1 = J | X 0 = I) for all I, J ⊆ S. Put differently, we ask the question under which conditions on the distribution p = (p K ) K⊆S the Markov transition matrix of X occurs in a time-homogeneous Markov semigroup of the form {e tQ : t 0} for some Markov generator Q. More generally, one can also consider the time-inhomogeneous situation of a general Markov flow; compare [13,5] and references therein. For our special matrix family, this extension does not seem to provide extra insight, wherefore we only touch upon it briefly.…”

“…Concrete and exhaustive criteria for a discrete-time Markov chain to be embeddable this way are known for state spaces of up to cardinality 4; see [6,4] and references therein. Beyond this, they are restricted to specific families of Markov matrices; see [3,5] for examples. The MCCP enriches this collection by an interesting member for which a complete, yet non-trivial answer can be given.…”