Some Probabilistic Properties of Bessel Functions

The Bessel function ratios (b/a)^ν K_ν(as1/2) (a > b > 0, ν ∈ R) and (b/a)^ν I_ν(as1/2)/I_ν(bs1/2) (0 < a < b, ν > -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.