when $k_t=t$ for all $t$ and to a standard PF algorithm when $k_t=1$ for all $t$. Notice that if $k_t=\Delta$ for some $\Delta\in\mathbb{N}$ then the proposed SMC algorithm is an online algorithm. Our main contribution is then to provide, for *any* sequence $(k_t)_{t\geq 1}$, a way to construct sequences $(\tilde{\pi}_t)_{t\geq 1}$ that converge to $\delta_{\{\theta_\star\}}$, under standard regularity conditions on the statistical model.

As we will argue in Section \[sub:h\_t\], a trade-off between statistical and computational efficiency should be considered when choosing the sequence $(\tilde{\pi}_t)_{t\geq 0}$: the cheaper it is to approximate this sequence by Monte Carlo the slower it converges to $\delta_{\{\theta_\star\}}$. Averaging is a standard technique used to improve the statistical and/or predictive properties of online procedures, as briefly recalled in Section \[sub:averaging\]. Following this idea, we propose in that section to base the inference on the mixture distribution $\bar{\pi}_t:=(t+1)^{-1}\sum_{s=0}^t\tilde{\pi}_s$ instead of using $\tilde{\pi}_t$ directly. We show that the distribution $ \bar{\pi}_t $ converges to the same distribution as $\tilde{\pi}_t$ and that its online Monte Carlo estimation can be obtained as a by product of that of $\tilde{\pi}_t$. The merit of $\bar{\pi}_t$ over $\tilde{\pi}_t$ comes from the fact that, empirically, the point estimate $\bar{\theta}_t:={\mathbb{E}}_{\bar{\pi}_t}[\theta]$ typically converges to $\theta_\star$ 