to $\theta_\star$ at rate $t^{-1/2}$ (see the numerical examples of Sections \[subsec:example\], \[sec:num\_quantile\] and \[sub:latent\]) while the convergence behaviour of $\tilde{\theta}_t$ depends on the speed at which $h_t\rightarrow 0$, as illustrated in the next subsection. We note that this property of estimators obtained by averaging online point estimates is well-known in the SA literature and has been proved in this context [@polyak1992acceleration]. On the other hand, and assuming $\Theta\subseteq{\mathbb{R}}$ to simplify the presentation, the inequalities $$\begin{aligned}
\label{eq:lower_var}
{\mathrm{Var}}_{\scalebox{1.5}{$\scriptscriptstyle \bar{\pi}_t$}}(\theta)\geq \frac{1}{t+1}\sum_{s=0}^t{\mathrm{Var}}_{\scalebox{1.5}{$\scriptscriptstyle \tilde{\pi}_s$}}(\theta)\geq \min_{s\leq t}{\mathrm{Var}}_{\scalebox{1.5}{$\scriptscriptstyle \tilde{\pi}_s$}}(\theta) ,\quad\forall t\geq 0,\quad {\mathbb{P}}-a.s.\end{aligned}$$ suggest that, in general, $\bar{\pi}_t$ will not concentrate on $\theta_\star$ at a faster rate than $\tilde{\pi}_t$. Indeed, if $\tilde{\pi}_t$ converges to $\delta_{\{\theta_\star\}}$ then, under mild conditions, ${\mathrm{Var}}_{ \tilde{\pi}_t}(\theta)\rightarrow 0$, in which case it is reasonable to expect that, in , we have $ \min_{s\leq t}{\mathrm{Var}}_{\tilde{\pi}_s}(\theta)\approx {\mathrm{Var}}_{\tilde{\pi}_t}(\theta )$.

A second advantage of $(\bar{\pi}_t)_{t\geq 0}$ over $(\tilde{\pi}_t)_{t\geq 0}$ is that, empirically, the evolution of the former sequence is more stable than that of latter. One way to formalize this observation is to remark that we can establish the almost sure convergence of $\bar{\pi}_t$ to $\delta_{\{\theta_\star\}}$ under weaker conditions than for $\tilde{\pi}_t$. For instance, as a direct consequence of @lyons1988strong [Corollary 4], to guarantee that 