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# Symmetry characterizations of certain distributions, 1

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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.

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1Bingham, N. H., Goldie, C. M. and Teugels, J. L.. Regular Variation, Encyclopaedia of Mathematics and its Applications, vol. 27 (Cambridge University Press, 1987).
2Bingham, N. H. and Rogers, L. C. G.. Summability methods and almost sure convergence. (To appear.)
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.
4Rogers, L. C. G. and Williams, D.. Diffusions, Markov Processes and Martingales, vol. 2: It calculus (Wiley, 1987).
5Titchmarsh, E. C.. The Theory of Functions, 2nd edition (Oxford University Press, 1939).
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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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