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Technical comment. A problem on Markov chains

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RAIRO Operations Research

RAIRO Oper. Res.36(2002) 173 DOI: 10.1051/ro:2002011

TECHNICAL COMMENT

A PROBLEM ON MARKOV CHAINS Franco Giannessi

1

Abstract. A problem (arisen from applications to networks) is posed about the principal minors of the matrix of transition probabilities of a Markov chain.

Keywords: Markov chains, stochastic matrices.

1. The problem

Let the Markov chain (P0, P) be given, where P is the n×n matrix, whose entries are the transition probabilities, andP0∈[0,1]nis the initial distribution [1].

Every K ⊆ {1, . . . , n} identifies a principal minor ofP, denoted by PK; let I|K|

be the identity matrix of order|K|. It is not difficult to show that 0≤det(I|K|−PK)≤1, ∀K⊆ {1. . . , n}

where det denotes determinant. Is det (I|K|−PK) a (conditioned or not) proba- bility of an event related to Markov chain?

Reference

[1] S.N. Ethier and T.G. Kurtz,Markov processes: Characterization and convergence. J. Wiley, New York (1986).

Received June, 2002.

1 Deptartment of Mathematics, University of Pisa, Via F. Buonarroti 2, 56127 Pisa, Italy;

e-mail:gianness@dm.unipi.it

c

EDP Sciences 2002

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