论文标题
Hartman-Watson分布的渐近扩展
Asymptotic expansion for the Hartman-Watson distribution
论文作者
论文摘要
具有密度$ f_r（t）$的Hartman-Watson分布是在$ t \ geq 0 $上定义的概率分布，它出现在几个应用概率的问题中。此分布的密度是用积分$θ（r，t）$表示的，对于小$ t \至0 $，很难用数值评估。使用鞍点方法，我们在固定$ρ> 0 $下获得$ t \至0 $扩展为$θ（ρ/t，t）$的前两个项。通过对集成的数值估计来获得错误绑定，此外，该估计是$ρ$的均匀统一。作为应用程序，我们以$ t \ to $ t \ to $ t \ 0 $获得了几何布朗运动的时间平均密度的主要渐近学。它具有$ \ mathbb {p}（\ frac {1} {t} \ int_0^t e^{2（b_s+μs）} ds \ in da）\ sim（2πt）^{ - 1/2} g（b_s+μs）} ds^{ - 1/2} g（a，μ） $ j（a）$重现了先前使用大偏差理论获得的已知结果。
The Hartman-Watson distribution with density $f_r(t)$ is a probability distribution defined on $t \geq 0$ which appears in several problems of applied probability. The density of this distribution is expressed in terms of an integral $θ(r,t)$ which is difficult to evaluate numerically for small $t\to 0$. Using saddle point methods, we obtain the first two terms of the $t\to 0$ expansion of $θ(ρ/t,t)$ at fixed $ρ>0$. An error bound is obtained by numerical estimates of the integrand, which is furthermore uniform in $ρ$. As an application we obtain the leading asymptotics of the density of the time average of the geometric Brownian motion as $t\to 0$. This has the form $\mathbb{P}(\frac{1}{t} \int_0^t e^{2(B_s+μs)} ds \in da) \sim (2πt)^{-1/2} g(a,μ) e^{-\frac{1}{t} J(a)} da/a$, with an exponent $J(a)$ which reproduces the known result obtained previously using Large Deviations theory.
