## The Annals of Probability

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

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

Philip Hartman and Geoffrey S. Watson

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176996606

**Digital Object Identifier**

doi:10.1214/aop/1176996606

**Mathematical Reviews number (MathSciNet)**

MR370687

**Zentralblatt MATH identifier**

0305.60033

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 33A40

**Keywords**

Brownian diffusion Fisher distribution complete monotonicity modified Bessel functions spherical harmonics

#### Citation

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. http://projecteuclid.org/euclid.aop/1176996606.