# 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 $$mathbb{R}^n$$)?