Exact Tail Asymptotics of Dirichlet Distributions

Let X be a generalised symmetrised Dirichlet random vector in R^k, and let u_n be thresholds such that P{X> u_n} tends to 0 as n goes infinity. In this paper we derive an exact asymptotic expansion of P{X> u_n} assuming that the associated random radius of X has distribution function in the Gumbel max-domain of attraction