results to hold. Notice that under the rate at which $\|{\mathrm{Var}}_{\mu_{t}}(\theta)\|\rightarrow 0$ is determined by the sequence $(h_t)_{t\geq 0}$.

Then, for a given prior distribution $\tilde{\pi}_0\in\mathcal{P}_L(\Theta)$, we let $(\tilde{\pi}_t)_{t\geq 1}$ be defined by $$\begin{aligned}
\label{eq:pi_tilde}
\tilde{\pi}_t({\mathrm{d}}\theta)=\frac{ \tilde{f}_\theta(Y_t)(\mu_{t-1}*\tilde{\pi}_{t-1})({\mathrm{d}}\theta)}{\int_{{\mathbb{R}}^d} \tilde{f}_\theta(Y_t)(\mu_{t-1}*\tilde{\pi}_{t-1})({\mathrm{d}}\theta)}\in \mathcal{P}_L(\Theta) ,\quad  t\geq 1.\end{aligned}$$

In Section \[sec:main\] we provide conditions on $(h_t)_{t\geq 0}$ and on $(t_p)_{p\geq 0}$ that ensure the convergence of $\tilde{\pi}_t$ towards $\delta_{\{\theta_\star\}}$, under standard assumptions on the statistical model. For instance, our main results enable us to define, for every $\alpha>0$ and sequence $(h_t)_{t\geq 0}$ such that $h_t=\Theta(t^{-\alpha})$, a strictly increasing sequence $(t_p)_{p\geq 0}$ for which $\tilde{\pi}_t\Rightarrow \delta_{\{\theta_\star\}}$ in ${\mathbb{P}}$-probability.

The definition and of $(\tilde{\pi}_t)_{t\geq 0}$ guarantees that if $h_s=0$ for all $s<t$ then $\tilde{\pi}_t=\pi_t$, where we recall that $\pi_t$ is the posterior distribution of $\theta$. This observation suggests that the size of $\max_{0\leq s< t}h_s$ controls the discrepancy between $\tilde{\pi}_t$ and $\pi_t$. The following result supports this intuition and justifies the interpretation of $\tilde{\pi}_t$ as an approximation of $\pi_t$. Notice that for this result to hold it is not necessary to assume the $Y_t$’s are i.i.d.

\[prop:continuity\] Assume ${\mathbb{P}}(\sup_{\theta\in\Theta} f_\theta(Y_1)<\infty)=1$, that the mapping $\theta\mapsto f_\theta(Y_1)$ is ${\mathbb{P}}$-a.s. continuous on $\Theta$ and that $\tilde{\pi}_0\in\mathcal{P}_L(\Theta)$ has a continuous and bounded 