at the optimal $t^{-1/2}$ rate under various definitions of $(\tilde{\pi}_t)_{t\geq 0}$ while, on the contrary, the convergence behaviour of $\tilde{\theta}_t:={\mathbb{E}}_{\tilde{\pi}_t}[\theta]$ is very sensitive to the choice of this sequence, as we shall see in the numerical example of Section \[subsec:example\].

Establishing theoretically the convergence rate of the distributions $\tilde{\pi}_t$ and of $\bar{\pi}_t$, and of the point estimators $\tilde{\theta}_t$ and $\bar{\theta}_t$, is however beyond the scope of the paper. Instead, the objective of the paper is to derive consistency results for $(\tilde{\pi}_t)_{t\geq 0}$ and $(\bar{\pi}_t)_{t\geq 0}$ that hold under weak conditions on the statistical model at hand, and by imposing minimal constraints on the sequence $(\mu_t)_{t\geq 0}$. An unavoidable consequence (see Section \[subsec:example\]) of the very few conditions we impose on this sequence is that our convergence results for $(\tilde{\pi}_t)_{t\geq 0}$ hold *in probability*, while under the same assumptions on the model the posterior distribution $\pi_t$ converges to $\delta_{\{\theta_\star\}}$ *almost surely*. Our theoretical analysis carefully deals with the case where $\Theta$ is unbounded, which is particularly important since, in this case, nothing prevents the distributions $\tilde{\pi}_t$ and $\pi_t$ to be arbitrarily far from each other at a given time $t$.

Additional notation and outline of the paper\[sub:set-up\]
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We let $\|\cdot\|_\infty$ 