convergence results (Theorems \[thm:online\]-\[thm:lower\_bound\_1\]) are given in Proposition \[prop:h\_t\]. All the numerical experiments of Section \[sec:num\] rely on Algorithm \[algo:pi\_tilde\_online\] with $c=1$ and $\alpha=0.5$, except those of Section \[sub:linear\_model\].

For every $t\geq 0$, Algorithm \[algo:pi\_tilde\_online\] computes and approximation $\tilde{\pi}_t^N$ of $\tilde{\pi}_t$, where $N\in\mathbb{N}$ is the Monte Carlo effort. Using standard results for PF algorithms [see @crisan2000 Theorem 1, and Proposition \[prop:conv\_algo\] below], it can be shown that, as $N$ tends to $\infty$, the approximation $\tilde{\pi}_t^N$ converges to $\tilde{\pi}_t$ at the usual $N^{-1/2}$ Monte Carlo convergence rate. Notice that in Algorithm \[algo:pi\_tilde\_online\] each observation $Y_t$ is read only once and that the memory and computational requirement to process $Y_t$ does not increase with $t$ and is linear in $N$. A more precise description of Algorithm \[algo:pi\_tilde\_online\] is omitted, since it is a particular case of Algorithm \[algo:pi\_tilde\] which is introduced in details in Section \[sub:Monte\_Carlo\].

It is clear from Algorithm \[algo:pi\_tilde\_online\] that, for fixed $N$, the larger $\alpha$ is (i.e. the faster $h_t$ converges to $0$) the higher is the probability that the support of $\tilde{\pi}_t^N$ is close to the support of $\tilde{\pi}_{t-1}^N$. Hence, if $\alpha$ is large and the algorithm is initialized faraway from $\theta_\star$, a large number of iterations 