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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 $theta(r,t)$ which is difficult to evaluate numerically for small $tto 0$. Using saddle point methods, we obtain the first two terms of the $tto 0$ expansion of $theta(rho/t,t)$ at fixed $rho >0$. An error bound is obtained by numerical estimates of the integrand, which is furthermore uniform in $rho$. As an application we obtain the leading asymptotics of the density of the time average of the geometric Brownian motion as $tto 0$. This has the form $mathbb{P}(frac{1}{t} int_0^t e^{2(B_s+mu s)} ds in da) = (2pi t)^{-1/2} g(a,mu) e^{-frac{1}{t} J(a)} (1 + O(t))$, with an exponent $J(a)$ which reproduces the known result obtained previously using Large Deviations theory.
This article is concerned with the joint law of an integrated Wishart bridge process and the trace of an integrated inverse Wishart bridge process over the interval $ left[0,tright] $. Its Laplace transform is obtained by studying the Wishart bridge
It is well known that any positive matrix can be scaled to have prescribed row and column sums by multiplying its rows and columns by certain positive scaling factors (which are unique up to a positive scalar). This procedure is known as matrix scali
The two-parameter Poisson-Dirichlet distribution is the law of a sequence of decreasing nonnegative random variables with total sum one. It can be constructed from stable and Gamma subordinators with the two-parameters, $alpha$ and $theta$, correspon
This paper continues our treatment of the Neutron Transport Equation (NTE) building on the work in [arXiv:1809.00827v2], [arXiv:1810.01779v4] and [arXiv:1901.00220v3], which describes the flux of neutrons through inhomogeneous fissile medium. Our aim
A version of the saddle point method is developed, which allows one to describe exactly the asymptotic behavior of distribution densities of Levy driven stochastic integrals with deterministic kernels. Exact asymptotic behavior is established for (a)