A necessary condition for weak lumpability in finite Markov processes

Abstract

Under certain conditions, the state space of a homogeneous Markov process can be partitionned to construct an aggregated markovian process. However, the verification of these conditions requires expensive computations. In this note, we expose a necessary condition for having a markovian aggregated process. This condition is based on properties of the eigenvalues of certain submatrices of the transition rate matrix of the original Markov process.

A geometric invariant in weak lumpability of finite Markov chains

“…Since Remark 2. The fact that P(/, I) is necessarily an eigenvalue of PC(l)C(l) completely generalizes the result given in [6] for an irreducible original chain. It was based on the fact that the Markovian property induces geometric sojourn times in each class C(/) and also on the Jordan canonical form of a matrix.…”

“…After reviewing some preliminaries on polyhedral cones, we analyze in Section 2, for a general finite Markov chain with transition probability matrix P, the set of initial distributions which give aggregated Markov chains sharing the same t.p.m. Pointing out the relation between lumpability and positive invariance of cones in Section 3, we show that this set is non-empty if there exists a family of M polyhedral cones that are 'invariant' under sub-matrices of matrix P. This result allows us to state in Section 4 that if the partition f!JJ is a refinement of the partition of the state space S induced by the usual 'communication' equivalence relation, then we obtain an explicit formula for the transition probability matrix of any Y, which depends only on f}J and P. Throughout Sections 3 and 4, various properties reported in [2], [6], [9] are extended to general finite Markov chains and new spectral results are also derived.…”

“…The previous theorem can be interpreted as follows. If a state of a class I,is allowed to be an initial state of our Markovian model then all the rows of matrix P corresponding to the state classes of the f!/J included in I, or in the element of a path with starting point I b are necessarily given by formula(6) and depend only on J(&» and P.…”

“…This result allows us to state in Section 4 that if the partition P is a refinement of the partition of the state space S induced by the usual "communication" equivalence relation, then we obtain an explicit formula for the transition probability matrix of any Y , which depends only on P and P . Throughout Section 3 and Section 4, various properties reported in Ledoux [6], Abdel-Moneim and Leysieffer [2] and Peng [9] are extended to general finite Markov chains and new spectral results are also derived.…”

Weak lumpability and pseudo-stationarity of finite markov chains

“…Consequently, for every n ≥ 1, the distribution of the nth sojourn time in C(l) is Geo( P (l, l)) if v n ∈ G P C(l)C(l) ( P (l, l)). Starting with the PH-representation of distributions of sojourn times in each class C(l) and using the canonical Jordan form of the matrix P C(l)C(l) , it was shown in [7] that P (l, l) (for all l ∈Ŝ) is an eigenvalue of matrix P C(l)C(l) . A similar argument should lead to a proof of Theorem 5.1.…”