that $\Theta={\mathbb{R}}^d$. Let $(\mu_t)_{t\geq 0}$ be a sequence of probability distributions on ${\mathbb{R}}^d$ and $(\theta_t)_{t\geq 0}$ be such that $\theta_0 \sim \tilde{\pi}_0$ and such that $(\{Y_t,\theta_t\})_{t\geq 1}$ is a bivariate Markov chain verifying $$\begin{aligned}
\label{eq:state_space}
Y_t|\theta_t\sim f_{\theta_t}(y_t)\eta({\mathrm{d}}y_t),\quad \theta_t=\theta_{t-1}+U_{t-1},\quad U_{t-1}\sim \mu_{t-1},\quad t\geq 1.\end{aligned}$$ Then, for all $t\geq 1$, we define $\tilde{\pi}_t$ as the conditional distribution of $\theta_t$ given the observations $\{Y_s\}_{s=1}^t$. If all the $(\mu_{t})_{t \geq 0}$ are dominated by a common dominating measure, then defines a state-space model. In this case, $\tilde{\pi}_t$ is the corresponding filtering distribution at time $t$, and particle filter (PF) algorithms [@doucet2009tutorial] can be used to compute an online Monte Carlo approximation of $(\tilde{\pi}_t)_{t\geq 0}$, where the memory and computational requirement of each iteration is bounded by $C N$, with $N\in\mathbb{N}:=\{1,2,\dots\}$ the Monte Carlo effort and where in the rest of this section $C<\infty$ is a constant independent on $t$.

For situations where does not define a state-space model, we develop a sequential Monte Carlo (SMC) algorithm to approximate $(\tilde{\pi}_t)_{t\geq 1}$ whose memory and computational requirement at iteration $t\geq 1$ is a most $C k_t N$, with $k_t=t-\sup\{s<t:\mu_s\neq\delta_{\{0\}}\}$ if $\{s<t:\mu_s\neq\delta_{\{0\}}\}$ is not an empty set and $k_t=t$ otherwise. This SMC sampler reduces to the IBIS algorithm 