## The Annals of Probability

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

#### 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

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