Particle filters (PFs) is a class of Monte Carlo algorithms that propagate over time a set of $N\in\mathbb{N}$ particles which can be used to estimate, in an online fashion, the sequence of filtering distributions $(\hat{\eta}_t)_{t\geq 1}$ defined by a state-space model. Despite the popularity of PFs, the time evolution of their estimates does not appear to have been previously studied in the literature. Denoting by $(\hat{\eta}_t^N)_{t\geq 1}$ the PF estimate of $(\hat{\eta}_t)_{t\geq 1}$ and letting $\kappa\in (0,1)$, we first show that for any number of particles $N$ it holds that, with probability one, we have $\|\hat{\eta}_t^N- \hat{\eta}_t\|\geq \kappa$ for infinitely many $t\geq 1$, with $\|\cdot\|$ a measure of distance between probability distributions. Considering a simple filtering problem we then provide reassuring results concerning the ability of PFs to estimate jointly a finite set $\{\hat{\eta}_t\}_{t=1}^T$ of filtering distributions by studying $\P(\sup_{t\in\{1,\dots,T\}}\|\hat{\eta}_t^{N}-\hat{\eta}_t\|\geq \kappa)$. Finally, on the same toy filtering problem, we prove that sequential quasi-Monte Carlo, a randomized quasi-Monte Carlo version of PF algorithms, offers greater safety guarantees than PFs in the sense that, for this algorithm, it holds that $\lim_{N\rightarrow\infty}\sup_{t\geq 1}\|\hat{\eta}_t^N-\hat{\eta}_t\|=0$ with probability one.

翻译：粒子滤波器（PFs）是一类蒙特卡洛算法，其在时间上传播一组$N\in\mathbb{N}$个粒子，可用于在线估计由状态空间模型定义的滤波分布序列$(\hat{\eta}_t)_{t\geq 1}$。尽管粒子滤波器应用广泛，但其估计值的时间演化此前在文献中似乎尚未被研究。记$(\hat{\eta}_t^N)_{t\geq 1}$为$(\hat{\eta}_t)_{t\geq 1}$的粒子滤波器估计，并令$\kappa\in (0,1)$。我们首先证明，对于任意粒子数$N$，以概率一成立：存在无限多个$t\geq 1$使得$\|\hat{\eta}_t^N- \hat{\eta}_t\|\geq \kappa$，其中$\|\cdot\|$为概率分布间的一种距离度量。针对一个简单的滤波问题，我们随后通过研究$\P(\sup_{t\in\{1,\dots,T\}}\|\hat{\eta}_t^{N}-\hat{\eta}_t\|\geq \kappa)$，给出了关于粒子滤波器联合估计有限滤波分布集$\{\hat{\eta}_t\}_{t=1}^T$能力的可靠结果。最后，在同一示例滤波问题上，我们证明序列拟蒙特卡洛（一种随机化拟蒙特卡洛版本的粒子滤波器算法）比标准粒子滤波器具有更高的安全性保证，即对于该算法，以概率一成立$\lim_{N\rightarrow\infty}\sup_{t\geq 1}\|\hat{\eta}_t^N-\hat{\eta}_t\|=0$。