${\mathbb{P}}(\bar{\pi}_t\Rightarrow\delta_{\{\theta_\star\}})=1$ it is enough to have, for all $\epsilon>0$, a $\delta_\epsilon>0$ such that ${\mathbb{E}}[\tilde{\pi}_t(V_\epsilon)]={\mathcal{O}}(t^{-\delta_\epsilon})$. By contrast, proving that ${\mathbb{P}}(\tilde{\pi}_t\Rightarrow\delta_{\{\theta_\star\}})=1$ usually requires to show that, for all $\epsilon>0$, ${\mathbb{E}}[\tilde{\pi}_t(V_\epsilon)]={\mathcal{O}}(t^{-\delta_\epsilon})$ for some $\delta_\epsilon>1$.

An illustrative example {#subsec:example}
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For every $\theta\in\Theta:={\mathbb{R}}$ we let $f_\theta$ be the density of the $\mathcal{N}_1(\theta,1)$ distribution and, for some $\tilde{\theta}_0\in{\mathbb{R}}$ and $\tilde \sigma_0^2>0$, we let $\tilde{\pi}_0({\mathrm{d}}\theta)$ be the density of the $\mathcal{N}_1(\tilde{\theta}_0,\tilde{\sigma}_0^2)$ distribution. Remark that for this model our general assumptions on $(Y_t)_{t\geq 1}$ impose that ${\mathbb{E}}[Y_1^2]<\infty$, since otherwise $\theta_\star=\operatorname*{argmax}_{\theta\in\Theta}{\mathbb{E}}[\log  f_\theta(Y_1)]$ is not defined. However, in this subsection we relax this condition and assume only that ${\mathbb{E}}[|Y_1|]<\infty$. Notice that if $\theta_\star$ is well-defined then $\theta_\star={\mathbb{E}}[Y_1]$. In addition, to make the computation of $(\tilde{\pi}_t)_{t\geq 0}$ tractable, we also relax the condition that $(\mu_t)_{t\geq 0}$ contains infinity many Student’s $t$-distributions (imposed in Section \[sub:pi\_tilde\]) by letting, in , $m_t$ be the density of the $  \mathcal{N}_1(0,1)$ distribution for all $t\geq 0$.

Then, with $g:{\mathbb{R}}\rightarrow{\mathbb{R}}$ defined by $g(x)=x/(1+x)$, $x\in (0,\infty)$, it can be easily checked that $$\tilde{\pi}_t=\mathcal{N}_1(\tilde{\theta}_t,\tilde{\sigma}_t^2),\quad \tilde{\sigma}_t^2=g(\tilde{\sigma}^2_{t-1}+h^2_{t-1}),\quad \tilde{\theta}_t=\tilde{\sigma}_t^2 Y_t + (1-\tilde{\sigma}_t^2)\tilde{\theta}_{t-1},\quad t\geq 1.$$ Since $g$ is strictly increasing on $(0,\infty)$ it follows that $\tilde{\sigma}_t^2$ is a strictly increasing function of $h_s$, for all $s\in\{0,\dots,t-1\}$. The impact of $(h_t)_{t\geq 0}$ 