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

  • Martin Baxter (a1) and David Williams (a1)
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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.

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References
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[1]Baxter, M. W. and Williams, D.. Symmetry characterizations of certain distributions, 1. Math. Proc. Cambridge Philos. Soc. 111 (1992), 387397.
[2]Deuschel, J.-D. and Stroock, D. W.. Large Deviations (Academic Press, 1989).
[3]Eastham, M. S. P.. The Asymptotic Solution of Linear Differential Systems (Oxford University Press, 1989).
[4]Ellis, R. S.. Large deviations for a general class of random vectors. Ann. Probab. 12 (1984), 112.
[5]Itô, K. and McKean, H. P.. Diffusion Processes and their Sample Paths (Springer-Verlag, 1965).
[6]Kato, T.. A Short Introduction to Perturbation Theory for Linear Operators (Springer-Verlag, 1982).
[7]Rogers, L. C. G. and Williams, D.. Diffusions, Markov Processes, and Martingales, vol. 2: Itô Calculus (Wiley, 1987).
[8]Seneta, E.. Non-negative Matrices, an Introduction to Theory and Applications (Allen & Unwin, 1973).
[9]Varadhan, S. R. S.. Large Deviations and Applications (SIAM, 1984).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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