be the maximum norm on ${\mathbb{R}}^d$, $B_\epsilon(x)$ be the open ball of size $\epsilon>0$ around $x\in{\mathbb{R}}^d$ w.r.t. $\|\cdot\|$, the Euclidean norm on ${\mathbb{R}}^d$, $t_{d,\nu}(m,\Sigma)$ be the $d$-dimensional Student’t-distribution, with $\nu>0$ degrees of freedom, location vector $m$ and scale matrix $\Sigma$, and $\mathcal{N}_d(m,\Sigma)$ be the $d$-dimensional Gaussian distribution with mean $m$ and covariance matrix $\Sigma $. Recall that for two strictly positive sequences $(a_t)_{t\geq 1}$ and $(b_t)_{t\geq 1}$ the notation $a_t=\Theta(b_t)$ means that $\liminf_{t\rightarrow\infty} (a_t/b_t+b_t/a_t)>0$.

Below we use the shorthand ${\mathbb{E}}[g]$ for ${\mathbb{E}}[g(Y_1)]$, $\nu(h)$ for $\int_{\Theta}h(\theta)\nu({\mathrm{d}}\theta)$, ${\mathrm{Var}}_{\nu}(h)$ for $\nu(h h^T)-\nu(h) \nu(h)^T$. In particular, ${\mathbb{E}}_\nu[\theta]$ stands for $\int_{\Theta} \theta\nu({\mathrm{d}}\theta)$ and ${\mathrm{Var}}_{\nu}(\theta)$ for ${\mathbb{E}}_\nu[\theta \theta^T]-{\mathbb{E}}_\nu[\theta]{\mathbb{E}}_\nu[\theta]^T$. If $A$ is a Borelian set of ${\mathbb{R}}^d$, we denote by $\mathcal{B}(A)$ the Borel $\sigma$-algebra on $A$, by $\mathcal{P}(A)$ the set of probability measures on $(A,\mathcal{B}(A))$ and we let $\mathcal{P}_L(A)$ be the set of probability measures on $(A,\mathcal{B}(A))$ that are absolutely continuous w.r.t. ${\mathrm{d}}\theta$, the Lebesgue measure on ${\mathbb{R}}^d$. Moreover, for a sequence $(\nu_t)_{t \geq 0}$ of probability measures on ${\mathbb{R}}^d$, the notation $\nu_t \Rightarrow \nu$ means that the sequence $(\nu_t)_{t \geq 0}$ converges weakly to the probability measure $\nu\in \mathcal{P}({\mathbb{R}}^d)$.

We let $f_\star(y)\eta({\mathrm{d}}y)$ be the distribution of $Y_1$ and assume throughout the paper that $\Theta\in\mathcal{B}({\mathbb{R}}^d)$, with $\int_{\Theta}{\mathrm{d}}\theta>0$. 