pr.probability – Weak convergence of Dirichlet distributions to a “multi-Bernouilli” distribution


For a positive vector $alphainmathbb{R}^n$ ($ngeq 1$), denote by $text{Dir}(alpha)$ the Dirichlet distribution with parameter alpha. In terms of weak convergence, is it true that $limlimits_{varepsilonrightarrow 0^+}text{Dir}(varepsilonalpha)longrightarrow sumlimits_{i=1}^n alpha_i delta_{lbrace e_irbrace}$ (where $(e_i)_{1leq ileq n}$ is the canonical base of $mathbb{R}^n$)?