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Probability Functions which are Proportional to Characteristic Functions and the Infinite Divisibility of the von Mises Distribution

Published online by Cambridge University Press:  05 September 2017

Toby Lewis
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Abstract

Reciprocal pairs of continuous random variables on the line are considered, such that the density function of each is, to within a norming factor, the characteristic function of the other. The analogous reciprocal relationship between a discrete distribution on the line and a continuous distribution on the circle is also considered. A conjecture is made regarding infinite divisibility properties of such pairs of random variables. It is shown that the von Mises distribution is infinitely divisible for sufficiently small values of the concentration parameter.

Type
Part II — Probability Theory
Information
Journal of Applied Probability , Volume 12 , Issue S1 , 1975 , pp. 19 - 28
Copyright
Copyright © 1975 Applied Probability Trust 

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References

Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. II. John Wiley, New York.Google Scholar
Kendall, D. G. (1974) Pole-seeking Brownian motion and bird navigation. J. R. Statist. Soc. B 36, 365–417.Google Scholar
Lukacs, E. (1970) Characteristic Functions. 2nd edn. Griffin, London.Google Scholar
Mardia, K. V. (1972) Statistics of Directional Data. Academic Press, London.Google Scholar
Sharpe, M. (1969) Zeros of infinitely divisible densities. Ann. Math. Statist. 40, 1503–1505.CrossRefGoogle Scholar
Stephens, M. A. (1963) Random walk on a circle. Biometrika 50, 385–390.CrossRefGoogle Scholar