I denote $p, q$ as density function of $P, Q$. Given $Y, X$ are random variables and \begin{align} \int q(x)\mathbb{D}_{\alpha}[p(Y\mid X=x)\,||\,p(Y\mid x^\star)] \,dx \rightarrow 0 \end{align} where $\mathbb{D}_\alpha[\cdot]$ is Rényi's $\alpha$-divergence, i.e. $D_\alpha[P||Q] = \frac{1}{\alpha-1}\log\int p^\alpha(x) q^{1-\alpha}(x)\,dx$, $x^\star$ is a fix value. Can we prove $\mathbb{E}_{Q(X)}[X] \rightarrow x^\star$ and $\mathbb{V}\text{ar}_{Q(X)}[X] \rightarrow 0$?
I know (can prove) that $\mathbb{D}_{\alpha}[p\,||\,q] = 0$ iff $p=q$, but don't know how to apply it to prove the above claims.