For what follows it is convenient to extend the parameter space $\Theta$ to ${\mathbb{R}}^d$. To this aim, for every $\theta\in{\mathbb{R}}^d$ we let $\tilde{f}_\theta:{\mathsf{Y}}\rightarrow{\mathbb{R}}$ be such that $\tilde{f}_\theta\equiv f_\theta$ for all $\theta\in \Theta$ and such that $\tilde{f}_\theta\equiv 0$ for all $\theta \in {\mathbb{R}}^d\setminus \Theta$.

Lastly, for any probability measure $\nu$ on $\Theta$ and any sequence $(\nu_t)_{t \geq 0}$ of probability measures on $\Theta$ (implicitly indexed by random variables), we say that $\nu_t \Rightarrow \nu$ in ${\mathbb{P}}$-probability if the sequence of random variables $(\varrho(\nu_t,\nu))_{t \geq 0}$ converges in ${\mathbb{P}}$-probability to $0$, where $\varrho$ denotes the Prohorov distance between probability measures on ${\mathbb{R}}^d$. Notice that if $\nu_t \Rightarrow \nu$ in ${\mathbb{P}}$-probability then, for every continuous and bounded measurable function $\varphi:\Theta\rightarrow{\mathbb{R}}$, $\nu_t(\varphi)\rightarrow \nu(\varphi)$ in ${\mathbb{P}}$-probability. In the case where $\nu=\delta_{\{\theta_\star\}}$, and letting $V_\delta=\{\theta\in{\mathbb{R}}^d:\,\|\theta-\theta_\star\|\geq\delta\} $ for every $\delta>0$, @Ghosal2017 [Proposition 6.2 and above comment, page 124] provide the following necessary and sufficient condition for checking that $\nu_t \Rightarrow \nu$ in ${\mathbb{P}}$-probability.

\[prop:conv\_proba\] $\nu_t \Rightarrow \delta_{\{\theta_\star\}}$ in ${\mathbb{P}}$-probability if and only if ${\mathbb{E}}[\nu_t(V_\epsilon)]\rightarrow 0$ for all $\epsilon>0$.

The rest of the paper is organized as follows. The proposed approach is described in detail in Section \[sec:approach\] and convergence results for the sequences $(\tilde{\pi}_t)_{t\geq 0}$ and $(\bar{\pi}_t)_{t\geq 