the convergence rate of $\tilde{\sigma}^2_t$ and of $\bar{\sigma}_t^2$, but it turns out that the rate at which the estimator $\tilde{\theta}_t$ converges towards ${\mathbb{E}}[Y_1]$ is also impacted by the definition of this sequence. The numerical results in Table \[table:ex\] indeed show that $|\tilde{\theta}_t-{\mathbb{E}}[Y_1]|\approx t^{-\alpha/2}$ when $h_t=t^{-\alpha}$ for all $t\geq 1$. By contrast, we observe in Table \[table:ex\] that $\bar{\theta}_t$ converges to ${\mathbb{E}}[Y_1]$ at rate $t^{-1/2}$, regardless the value of $\alpha$. The numerical experiments of Sections \[sec:num\_quantile\] and \[sub:latent\] suggest that, in general, this property of $\bar{\theta}_t$ also holds true for more complicated models.

![Evolution of $d_{\mathrm{MMD}}(\tilde{\pi}_t,\pi_t)$ (left plot) and of a Monte Carlo estimate of $d_{\mathrm{MMD}}(\bar{\pi}_t,\pi_t)$ (right plot) for the example of Section \[subsec:example\]. The observations, $\tilde{\pi}_0$ and $h_0$ are as in Table \[table:ex\], while $h_t=t^{-\alpha}$ for all $t\geq 1$, with $\alpha=0.5$ (solid line), $\alpha=0.7$ (dashed line) and $\alpha=1$ (dotted line). Note that only the results for $\alpha\in\{0.5,1\}$ are presented for $d_{\mathrm{MMD}}(\tilde{\pi}_t,\pi_t)$. []{data-label="fig:toy_mmd"}](figs/toy_example_mmd.pdf "fig:")![Evolution of $d_{\mathrm{MMD}}(\tilde{\pi}_t,\pi_t)$ (left plot) and of a Monte Carlo estimate of $d_{\mathrm{MMD}}(\bar{\pi}_t,\pi_t)$ (right plot) for the example of Section \[subsec:example\]. The observations, $\tilde{\pi}_0$ and $h_0$ are as in Table \[table:ex\], while $h_t=t^{-\alpha}$ for all $t\geq 1$, with $\alpha=0.5$ (solid line), $\alpha=0.7$ (dashed line) and $\alpha=1$ (dotted line). 