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.

中文翻译：

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Hartman-Watson分布的小规模扩张

与密度的哈特曼沃森分布\（F_ {R}（T）= \压裂{1} {I_ {0}（R）} \ THETA（R，T）\）与[R > 0是定义的一个概率分布\（t \ in \ mathbb {R} _ {+} \），这出现在应用概率的几个问题中。该分布的密度由一个积分θ（r，t）给出，对于小t →0很难进行数值评估。使用鞍点方法，我们获得θ（ρ /）的t →0展开的前两项。t，t）固定ρ > 0。