Let $(X_n)_{n\geq 1}$ be a sequence of random variables defined on the $d-$simplex ($d\geq 1$) : $\Sigma_d=\big\lbrace\boldsymbol{x}\in\mathbb{R}_+^d,\,\sum_{1\leq i\leq n} x_i=1\big\rbrace$. Assuming that there exists $\alpha\in\Sigma_d$ such that for $n\geq 1$, $\mathbb{E}[X_n]=\alpha$, and that the sequence of covariance matrices $\text{Cov}(X_n)$ converges to the matrix with coefficients $M_{ij}=\alpha_i(\delta_{ij}-\alpha_j)$ ($i,j\in \{1,...,d\}$ and $\delta_{ij}$ is the Kronecker symbol), does the following weak convergence holds: $$\lim\limits_{n\rightarrow\infty}\,\mathbb{P}_{X_n}\longrightarrow \sum\limits_{1\leq i\leq d} \alpha_i \delta_{e_i},$$ where $(e_i)_{1\leq i\leq d}$ is the canonical base of $\mathbb{R}^d$, and $\delta_{x}$ the Dirac measure for $x\in\mathbb{R}^d$?

The specific case for which every $X_n$ follow a Dirichlet law was solved: Weak convergence of Dirichlet distributions to a "multi-Bernoulli" distribution .