the same rate. The numerical experiments of Section \[sec:num\] suggest that, when $h_t=t^{-\alpha}$ for all $t\geq 1$, having a convergence rate for the variance of $\theta$ under $\bar{\pi}_t$ which is slower than that of $h_t^2$ is not specific to the model we consider in this subsection but generally holds true.

                               $z_t\backslash\alpha$                               0.1    0.3    0.5    0.7     1
  ------------------------------------------------------------------------------- ------ ------ ------ ------ ------
   ${\mathrm{Var}}_{\scalebox{1.5}{$\scriptscriptstyle \tilde{\pi}_t$}}(\theta)$   0.09   0.30   0.50   0.70   1.00
    ${\mathrm{Var}}_{\scalebox{1.5}{$\scriptscriptstyle \bar{\pi}_t$}}(\theta)$    0.09   0.30   0.5    0.7    0.93
                      $|\tilde{\theta}_t-{\mathbb{E}}[Y_1]|$                       0.05   0.15   0.24   0.36   0.49
                       $|\bar{\theta}_t-{\mathbb{E}}[Y_1]|$                        0.52   0.52   0.52   0.53   0.68

  : Ordinary least square estimate of $\beta_2$ for the model $\log(z_t)=\beta_1-\beta_2\log(t)+\epsilon_t$ and for the example of Section \[subsec:example\]. The results are for $T=10^7$ observations, $(\tilde{\theta}_0,\tilde{\sigma}^2_0)=(0,25)$, $Y_t{\stackrel{\mathclap{\normalfont\mbox{\tiny{iid}}}}{\sim}}\mathcal{N}_1(0,1)$, $h_0=0$ and $h_t=t^{-\alpha}$ for all $t>1$. \[table:ex\]

The first part of the proposition shows that the condition $h_t\rightarrow 0$ is enough to guarantee the convergence in ${\mathbb{P}}$-probability of $\tilde{\pi}_t$ towards $\delta_{{\mathbb{E}}[Y_1]}$. Our main results (Theorems \[thm:online\]-\[thm:lower\_bound\_1\]) show that weak conditions on the sequence $(h_t)_{t\geq 0}$ are indeed needed to guarantee that $\tilde{\pi}_t\Rightarrow\delta_{\{\theta_\star\}}$ in ${\mathbb{P}}$-probability. However, for reasons explained in Section \[sub:student\], these results require that infinitely many elements of the sequence $(\mu_t)_{t\geq 0}$ are Student’s t-distributions.

We saw above how the choice of $(h_t)_{t\geq 0}$ affects 