Some Probabilistic Properties of Bessel Functions
John T. Kent · The Annals of Probability · 1978
The Bessel function ratios $(b/a)^ u K_ u(as^{\frac{1}{2}}) (a > b > 0, u \in R)$ and $(b/a)^ u I_ u(as^{\frac{1}{2}})/I_ u(bs^{\frac{1}{2}}) (0 -1)$ 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.