Exact Tail Asymptotics of Dirichlet Distributions

Abstract

Let $\X=A^\top R\U$ be a linearly transformed generalised symmetrised Dirichlet scale mixture in $\R^k$, $k\ge2$. For a fixed direction $\b\in(0,\infty)^k$, we derive an exact asymptotic expansion of $\pk{\X>\vk t_n}$ for eventually positive threshold vectors $\vk t_n$ described relative to the quadratic-programming minimiser on the natural active and residual Gumbel scales; residual limits equal to $-\infty$ are allowed. The radial distribution is assumed to belong to the Gumbel max-domain of attraction. The local power and constant are determined by the local product-power behaviour of the angular density near the minimising direction. The result includes the ray $\vk t_n=u_n\b$ and yields an explicit comparison with the associated elliptical model, a conditional weak limit for the locally rescaled vector and the limiting location of the smallest component under a high common threshold. The minimum overshoot is asymptotically exponential and independent of its location. The finite-dimensional Gaussian minimum and location limits are recovered as a special case.