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Symmetry characterizations of certain distributions. 3. Discounted additive functional and large deviations for a general finite-state Markov chain

Published online by Cambridge University Press:  24 October 2008

Martin Baxter
Martin Baxter
Affiliation:
Statistical Laboratory, University of Cambridge., Cambridge CB2 1SB
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Extract

In Baxter and Williams [1] we began a study of Abel averages,

as opposed to the oft-studied Cesàro averages In Baxter and Williams [2], hereinafter referred to as [BW2], we studied the large-deviation behaviour of these averages. In the case where X is an irreducible Markov chain on a finite state-space S = {1,…, n}, we observed that

and

where π is the invariant distribution of X. We noted that

where v is an n-vector, δ(v):= sup{Re(z): z ∈ spect(Q+ V)}, and where spect(·) denotes spectrum (here the set of eigenvalues), Q is the Q-matrix of X, and V denotes the diagonal matrix diag (vi). It is also true that the large-deviation property holds for Ct with rate function I denned on M = {(xt)i ∈ S: xi ≥ 0, Σixi = 1}.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 2 , March 1993 , pp. 381 - 386
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Baxter, M. W. and Williams, D.. Symmetry characterizations of certain distributions, 1. Math. Proc. Cambridge Philos. Soc. 111 (1992), 387–397.CrossRefGoogle Scholar
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[5]Seneta, E.. Non-negative Matrices, an Introduction to Theory and Applications (Allen and Unwin, 1973).Google Scholar