For a positive vector $\alpha\in\mathbb{R}^n$ ($n\geq 1$), denote by $\text{Dir}(\alpha)$ the Dirichlet distribution with parameter $\alpha$. In terms of weak convergence, is it true that $\lim\limits_{\varepsilon\rightarrow 0^+}\text{Dir}(\varepsilon\alpha)\longrightarrow \sum\limits_{i=1}^n \alpha_i \delta_{\lbrace e_i\rbrace}$ (where $(e_i)_{1\leq i\leq n}$ is the canonical base of $\mathbb{R}^n$)?

For a positive vector $\alpha\in\mathbb{R}^n$ ($n\geq 1$), denote by $\text{Dir}(\alpha)$ the Dirichlet distribution with parameter $\alpha$. In terms of weak convergence, is it true that, if $\sum\limits_{i=1}^n\alpha_i=1$, then $\lim\limits_{\varepsilon\rightarrow 0^+}\text{Dir}(\varepsilon\alpha)\longrightarrow \sum\limits_{i=1}^n \alpha_i \delta_{\lbrace e_i\rbrace}$ (where $(e_i)_{1\leq i\leq n}$ is the canonical base of $\mathbb{R}^n$)?

For a positive vector $\alpha\in\mathbb{R}^n$ ($n\geq 1$), denote by $\text{Dir}(\alpha)$ the Dirichlet distribution with parameter alpha. In terms of weak convergence, is it true that $\lim\limits_{\varepsilon\rightarrow 0^+}\text{Dir}(\varepsilon\alpha)\longrightarrow \sum\limits_{i=1}^n \alpha_i \delta_{\lbrace e_i\rbrace}$ (where $(e_i)_{1\leq i\leq n}$ is the canonical base of $\mathbb{R}^n$)?

For a positive vector $\alpha\in\mathbb{R}^n$ ($n\geq 1$), denote by $\text{Dir}(\alpha)$ the Dirichlet distribution with parameter $\alpha$. In terms of weak convergence, is it true that $\lim\limits_{\varepsilon\rightarrow 0^+}\text{Dir}(\varepsilon\alpha)\longrightarrow \sum\limits_{i=1}^n \alpha_i \delta_{\lbrace e_i\rbrace}$ (where $(e_i)_{1\leq i\leq n}$ is the canonical base of $\mathbb{R}^n$)?