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Symmetry characterizations of certain distributions, 2: Discounted additive functionals and large deviations

Published online by Cambridge University Press:  24 October 2008

Martin Baxter  and
David Williams
Martin Baxter
Affiliation:
Statistical Laboratory, University of Cambridge, Cambridge CB2 1SB
David Williams
Affiliation:
Statistical Laboratory, University of Cambridge, Cambridge CB2 1SB
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Extract

This paper may be read independently of Baxter and Williams [1], hereinafter denoted by [BW1].

As in [BW1], we are mainly concerned with discounted additive functionals. We find that the large-deviation behaviour of the average depends on the precise average used. We derive, in certain cases, a link (but not equality) between the Cesàro average and Abel average limits, and would expect that other averages would produce other limiting behaviours. We have focused on the exponentially discounted (Abel average) case, both because of its tractability and because of its frequent appearance in decision/control problems and models of financial markets. We do give in subsection (e) the promised ‘excursion’ treatment of symmetry characterizations of the type announced in [BW1]; and this new treatment is simpler, more illuminating and more general. First, however, we focus attention on a different kind of asymptotic behaviour from that studied in [BW1], and on differential equations for exact results.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 3 , November 1992 , pp. 599 - 611
Copyright
Copyright © Cambridge Philosophical Society 1992

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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
[2]Deuschel, J.-D. and Stroock, D. W.. Large Deviations (Academic Press, 1989).Google Scholar
[3]Eastham, M. S. P.. The Asymptotic Solution of Linear Differential Systems (Oxford University Press, 1989).Google Scholar
[4]Ellis, R. S.. Large deviations for a general class of random vectors. Ann. Probab. 12 (1984), 1–12.CrossRefGoogle Scholar
[5]Itô, K. and McKean, H. P.. Diffusion Processes and their Sample Paths (Springer-Verlag, 1965).Google Scholar
[6]Kato, T.. A Short Introduction to Perturbation Theory for Linear Operators (Springer-Verlag, 1982).CrossRefGoogle Scholar
[7]Rogers, L. C. G. and Williams, D.. Diffusions, Markov Processes, and Martingales, vol. 2: Itô Calculus (Wiley, 1987).Google Scholar
[8]Seneta, E.. Non-negative Matrices, an Introduction to Theory and Applications (Allen & Unwin, 1973).Google Scholar
[9]Varadhan, S. R. S.. Large Deviations and Applications (SIAM, 1984).CrossRefGoogle Scholar