Longest Runs in Markov Chains

Norbert Kusolitsch
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Norbert Kusolitsch: Technische Universität Wien, Institut für Statistik und Wahrscheinlichkeitstheorie

A chapter in Probability and Statistical Inference, 1982, pp 223-230 from Springer

Abstract: Abstract Let R be a proper subset of the statespace S of a finite, homogeneous irreducible Markov-chain. Runs are defined as subsequences consisting of states from R. The paper investigates the length of the longest run. Asymptotic properties of the longest run are discussed and it is shown that the ratio of the length of the longest run and the logarithm of the total number of observations converges to -1/log λ a.s. where λ denotes the largest eigenvalue of the matrix Π of transition probabilities from R to R. This is the generalization of the similar result for the case of i i d observations, which was first treated by P.Erdös-A.Rényi [1].

Date: 1982
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-009-7840-9_20

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DOI: 10.1007/978-94-009-7840-9_20

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