Small-t Expansion for the Hartman-Watson Distribution

The Hartman-Watson distribution with density fr(t)=1I0(r)θ(r,t) with r > 0 is a probability distribution defined on t∈ ℝ₊, which appears in several problems of applied probability. The density of this distribution is given by an integral θ(r, t) which is difficult to evaluate numerically for small t → 0. Using saddle point methods, we obtain the first two terms of the t → 0 expansion of θ(ρ/t, t) at fixed ρ > 0.