of a state-space model. In Section \[sub:h\_t\] we comment on the trade-off between computational and statistical efficiency one should consider when defining $(\tilde{\pi}_t)_{t\geq 0}$, and make some practical recommendations. The sequential Monte Carlo algorithm for approximating an arbitrary sequence $(\tilde{\pi}_t)_{t\geq 0}$ and associated sequence $(\bar{\pi}_t)_{t\geq 0}$ is introduced in Section \[sub:Monte\_Carlo\]. Lastly, in Section \[sub:post\_app\] we briefly discuss the approximation error we make when the output of this algorithm is interpreted as an approximation of the posterior distribution $\pi_t$.

The sequence Lg {#sub:pi_tilde}
---------------

Let $(\mu_t)_{t\geq 0}$ be a sequence in $\mathcal{P}({\mathbb{R}}^d)$ such that $$\begin{aligned}
\label{eq:mu_def}
\mu_t({\mathrm{d}}\theta)=
\begin{cases}
h_t^{-d}m_t(\theta/h_t){\mathrm{d}}\theta, &h_t>0\\
\delta_{\{0\}}({\mathrm{d}}\theta),&h_t=0
\end{cases},\qquad\forall t\geq 0\end{aligned}$$ where $(h_t)_{t\geq 0}$ is a sequence in $[0,\infty)$ and where, for some strictly increasing sequence $(t_p)_{p\geq 0}$ in $\mathbb{N}_0:=\{0\}\cup\mathbb{N}$, $m_t$ is the p.d.f. of the $\mathcal{N}_d(0,\Sigma_t)$ distribution if $t\not\in (t_p)_{p\geq 0}$ and the p.d.f. of the $t_{d,\nu}(0,\Sigma_t)$ distribution otherwise, with $\nu>1$ and $(\Sigma_t)_{t\geq 0}$ a sequence of symmetric positive definite matrices for which $$\begin{aligned}
\label{eq:Sigma_t}
\exists c\in(0,\infty)\quad\text{such that }
\quad  \|\Sigma_t\|\vee  \|\Sigma^{-1}_t\| \leq c,\quad \forall t\geq 0.
 \end{aligned}$$ Explanations why we require that the sequence $(\mu_t)_{t\geq 0}$ contains infinitely many Student’s t-distributions are postponed to Section \[sub:student\]. Conditions $\nu>1$ and are also needed for our theoretical 