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August, 1974 "Normal" Distribution Functions on Spheres and the Modified Bessel Functions
Philip Hartman, Geoffrey S. Watson
Ann. Probab. 2(4): 593-607 (August, 1974). DOI: 10.1214/aop/1176996606


In $R^n$, Brownian diffusion leads to the normal or Gaussian distribution. On the sphere $S^n$, diffusion does not lead to the Fisher distribution which often plays the role of the normal distribution on $S^n$. On the circle $(S^1)$ and sphere $(S^2)$, they are known to be numerically close. It is shown that there exists a random stopping time for the diffusion which leads to the Fisher distribution. This follows from the fact, proved here, that the modified Bessel function $I_v(x)$ is a completely monotone function of $v^2$ (for fixed $x > 0$). More generally, we study the class of distributions on $S^n$ which can be represented as mixtures of diffusions. The stopping time distribution is characterized, but not given in computable form. Also, three new distribution functions involving Bessel functions are presented.


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Philip Hartman. Geoffrey S. Watson. ""Normal" Distribution Functions on Spheres and the Modified Bessel Functions." Ann. Probab. 2 (4) 593 - 607, August, 1974.


Published: August, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0305.60033
MathSciNet: MR370687
Digital Object Identifier: 10.1214/aop/1176996606

Primary: 60G40
Secondary: 33A40

Keywords: Brownian diffusion , complete monotonicity , Fisher distribution , modified Bessel functions , Spherical harmonics

Rights: Copyright © 1974 Institute of Mathematical Statistics

Vol.2 • No. 4 • August, 1974
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