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.
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I am grateful to Lingjiong Zhu for very useful discussions and comments.
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Appendix: Subleading correction
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
where
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
As \(\rho \to \infty \) we get, using the asymptotics of y1 → 0 from the proof of Prop. 2(ii),
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
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Pirjol, D. Small-t Expansion for the Hartman-Watson Distribution. Methodol Comput Appl Probab 23, 1537–1549 (2021). https://doi.org/10.1007/s11009-020-09827-5
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DOI: https://doi.org/10.1007/s11009-020-09827-5