The Annals of Probability

Some Probabilistic Properties of Bessel Functions

John Kent

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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.


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