Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T03:17:34.427Z Has data issue: false hasContentIssue false
  • English
  • Français

Symmetry characterizations of certain distributions, 1

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
Get access Rights & Permissions [Opens in a new window]

Extract

Certain interesting distributions are characterized by two or more symmetry properties, and it is a challenge to link these properties in an effective way. Dynkin's advice Discuss the simplest case first is always the best, and here we follow it (and avoid technical jargon) by discussing in detail just one concrete example, which concerns a discounted version of the arc-sine law. Theorem 1 B gives the symmetry properties for this case, and Theorem 1 A shows how they may be exploited to obtain an asymptotic result which we find rather surprising. Section 5 points to one of the generalizations which we shall examine in a sequel, and describes the natural formulation of the strange symmetry property which we first meet at (16b). (The sequel will be in many ways very different from this paper, but will share with it the fact that its results are not those which we originally conjectured.) In Sections 6 and 7 we consider some of the bizarre numerical analysis associated with the current problem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 2 , March 1992 , pp. 387 - 397
Copyright
Copyright © Cambridge Philosophical Society 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1Bingham, N. H., Goldie, C. M. and Teugels, J. L.. Regular Variation, Encyclopaedia of Mathematics and its Applications, vol. 27 (Cambridge University Press, 1987).CrossRefGoogle Scholar
2Bingham, N. H. and Rogers, L. C. G.. Summability methods and almost sure convergence. (To appear.)Google Scholar
3Hardy, G. H.. On the zeroes of certain classes of integral Taylor series. Part II: On the integral function and other similar functions. Proc. London Math. Soc. (2), 2, 401431, 1905.CrossRefGoogle Scholar
4Rogers, L. C. G. and Williams, D.. Diffusions, Markov Processes and Martingales, vol. 2: It calculus (Wiley, 1987).Google Scholar
5Titchmarsh, E. C.. The Theory of Functions, 2nd edition (Oxford University Press, 1939).Google Scholar
6Williams, D.. Diffusions, Markov Processes and Martingales, vol. 1: Foundations (Wiley, 1979).Google Scholar
7Williams, D.. Probability with Martingales (Cambridge University Press, 1991).CrossRefGoogle Scholar