Markov chain, non-decomposable

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2020 Mathematics Subject Classification: Primary: 60J10 Secondary: 60J27 [MSN][ZBL]

A Markov chain whose transition probabilities $P_{ij}(t)$ have the following property: For any states $i$ and $j$ there is a time $t_{ij}$ such that $p_{ij}(t_{ij}) > 0 $. The non-decomposability of a Markov chain is equivalent to non-decomposability of its matrix of transition probabilities $P = \left( {p_{ij}} \right)$ for a discrete-time Markov chain, and of its matrix of transition probability densities $Q = \left( {p'_{ij}(0)} \right)$ for a continuous-time Markov chain. The state space of a non-decomposable Markov chain consists of one class of communicating states (cf. Markov chain).


Cf. also Markov chain and Markov chain, decomposable for references.

How to Cite This Entry:
Markov chain, non-decomposable. Encyclopedia of Mathematics. URL:,_non-decomposable&oldid=39529
This article was adapted from an original article by B.A. Sevast'yanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article