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ISIPTA'07 -
FIFTH INTERNATIONAL SYMPOSIUM ON

IMPRECISE PROBABILITY: THEORIES AND APPLICATIONS

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Charles University, Faculty of Mathematicsand Physics

Prague, Czech Republic

16-19 July 2007

## ELECTRONIC PROCEEDINGS

## Damjan Škulj

# Regular finite Markov chains with interval probabilities

### Abstract

In Markov chain theory a stochastic matrix $P$ is regular if some matrix power $P^n$ contains only strictly positive elements. Regularity of transition matrix of a Markov chain guarantees the existence of a unique invariant distribution which is also the limiting distribution. In the present paper a similar result is shown for the generalized Markov chain model that replaces classical probabilities with interval probabilities. We generalize the concept of regularity and show that for a regular interval transition matrix sets of probabilities corresponding to consecutive steps of a Markov chain converge to a unique limiting set of distributions that only depends on transition matrix and is independent of the initial distribution. A similar convergence result is also shown for approximations of the invariant set.

** Keywords. ** Markov chains, interval probabilities

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Kardeljeva ploscad 5

1000 Ljubljana

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