## The Annals of Probability

- Ann. Probab.
- Volume 6, Number 5 (1978), 760-770.

### Some Probabilistic Properties of Bessel Functions

#### Abstract

The Bessel function ratios $(b/a)^\nu K_\nu(as^{\frac{1}{2}}) (a > b > 0, \nu \in R)$ and $(b/a)^\nu I_\nu(as^{\frac{1}{2}})/I_\nu(bs^{\frac{1}{2}}) (0 < a < b, \nu > -1)$ are infinitely divisible Laplace transforms in $s > 0$. These results are derived as hitting times of the Bessel diffusion process. The infinite divisibility of the $t$-distribution is deduced as a limiting result. A relationship with the von Mises-Fisher distribution is also demonstrated.

#### Article information

**Source**

Ann. Probab. Volume 6, Number 5 (1978), 760-770.

**Dates**

First available: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176995427

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aop/1176995427

**Mathematical Reviews number (MathSciNet)**

MR501378

**Zentralblatt MATH identifier**

0402.60080

**Subjects**

Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

Secondary: 33A40

**Keywords**

Infinite divisibility $t$-distribution Laplace transform Bessel functions diffusion semigroup von Mises-Fisher distribution

#### Citation

Kent, John. Some Probabilistic Properties of Bessel Functions. The Annals of Probability 6 (1978), no. 5, 760--770. doi:10.1214/aop/1176995427. http://projecteuclid.org/euclid.aop/1176995427.