The Annals of Probability

"Normal" Distribution Functions on Spheres and the Modified Bessel Functions

Philip Hartman and Geoffrey S. Watson

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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.

Article information

Ann. Probab. Volume 2, Number 4 (1974), 593-607.

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 33A40

Brownian diffusion Fisher distribution complete monotonicity modified Bessel functions spherical harmonics


Hartman, Philip; Watson, Geoffrey S. "Normal" Distribution Functions on Spheres and the Modified Bessel Functions. Ann. Probab. 2 (1974), no. 4, 593--607. doi:10.1214/aop/1176996606.

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