Small-t Expansion for the Hartman-Watson Distribution

Abstract

The Hartman-Watson distribution with density $f_{r}(t)=\frac {1}{I_{0}(r)} \theta (r,t)$ with r > 0 is a probability distribution defined on $t \in \mathbb {R}_{+}$, 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.

Appendix: Subleading correction

We give here the subleading correction to the asymptotic expansion of the Hartman-Watson integral θ(ρ/t, t) in Proposition 1.

The first two terms of the asymptotic expansion of the Hartman-Watson integral as t → 0 are

$$ \theta(\rho/t,t) = \frac{1}{2\pi t} G(\rho) e^{-\frac{1}{t}(F(\rho) - \frac{\pi^{2}}{2})} \left( 1 + \frac12 t \tilde g_{2}(\rho) + O(t^{2}) \right) $$

(56)

where

$$ \begin{array}{@{}rcl@{}} \tilde g_{2}(\rho) = \left\{ \begin{array}{cc} \frac{-12 + 9 \rho \cosh x_{1} - 2 \rho^{2} \cosh^{2} x_{1}+5\rho^{2}} {12(\rho \cosh x_{1} - 1)^{3}} & , 0 < \rho \leq 1 \\ \frac{12 + 9 \rho \cos y_{1} + 2 \rho^{2} \cos^{2} y_{1}-5\rho^{2}} {12(1+\rho \cos y_{1})^{3}} & , \rho > 1 \end{array} \right. \end{array} $$

(57)

The result follows straighforwardly by keeping the next terms in the expansions of the integrals appearing in the proof of Proposition 1.

The plot of $\tilde g_{2}(\rho )$ is given in the left panel of Fig. 6.

We give also a few properties of the function $\tilde g_{2}(\rho )$. This has an expansion around ρ = 1

$$ \tilde g_{2}(\rho) = - \frac{1}{35} + \frac{144}{67375} (1/\rho-1) + O((1/\rho-1)^{2}) . $$

(58)

As $\rho \to \infty$ we get, using the asymptotics of y₁ → 0 from the proof of Prop. 2(ii),

$$ \tilde g_{2}(\rho) = - \frac{1}{4\rho} + \frac{3}{2\rho^{2}} + O(\rho^{-3}) . $$

(59)

For ρ → 0, recall from the proof of Proposition 2 (i) that $\rho \cosh x_{1} \sim \log (1/\rho )\to \infty$, which gives the asymptotics

$$ \tilde g_{2}(\rho) = -\frac16 \frac{1}{\log(1/\rho)} + O(\log^{-2}(1/\rho)) . $$

(60)