Theoretical properties of particle approximations

All convergence results below involve taking limits as $N\rightarrow\infty$. Convergence almost surely is denoted $\overset{a.s.}{\rightarrow}$, convergence in probability is denoted $\overset{P}{\rightarrow}$ and weak convergence or convergence in distribution/law is denoted $\overset{L}{\rightarrow}$.

The following Theorem provides some justification for the use of the main particle approximations; these now classical results can be deduced from various results of Del Moral (2004). We present the results only for the "hatted" or "updated" quantities to avoid repetition; analogous hold for their "unhatted" counterparts with a different sequence of maps $\sigma_1^2, \ldots, \sigma_n^2$.

Theorem [Del Moral, 2004]. Let the potential functions $G_1, \ldots, G_n$ be bounded and strictly positive. There exist maps $\hat{\sigma}_1^2, \ldots, \hat{\sigma}_n^2$ such that the following hold for an arbitrary, bounded $f$:

• Lack-of-bias: $\mathbb{E} \left[ \hat{\gamma}_p^N(f) \right] = \hat{\gamma}_p(f)$ for all $N \geq 1$.

• Consistency: $\hat{\gamma}_p^N(f)\overset{a.s.}{\rightarrow}\hat{\gamma}_p(f)$ and $\hat{\eta}_p^N(f)\overset{a.s.}{\rightarrow}\hat{\eta}_p(f)$.

• Asymptotic variance and mean-squared error (MSE):

\[N {\rm var} \left \{ \hat{\gamma}_p^N(f) / \hat{\gamma}_p(1) \right \} \rightarrow \hat{\sigma}_p^2(f),\]

and

\[N \mathbb{E} \left[ \left\{ \hat{\eta}_p^N(f) - \hat{\eta}_p(f) \right\} ^2 \right] \rightarrow \hat{\sigma}_p^2(f-\hat{\eta}_p(f)).\]

• Central Limit Theorems:

\[\sqrt{N} \left( \hat{\gamma}_p^N(f) / \hat{\gamma}_p(1) - \eta_p^N(f) \right) \overset{L}{\rightarrow} N(0,\hat{\sigma}_p^2(f)),\]

and

\[\sqrt{N} \left( \hat{\eta}_p^N(f) - \eta_p^N(f) \right) \overset{L}{\rightarrow} N(0,\hat{\sigma}_p^2(f - \hat{\eta}_p(f))).\]

Note on the sorted ancestor indices

The theoretical results above are typically proven for an algorithm that differs very slightly from the SMC algorithm implemented here. In particular, one would usually analyze the algorithm by considering $A_{p-1}^1,\ldots,A_{p-1}^N$ to be i.i.d. ${\rm Categorical} (G_{p-1}(\zeta_{p-1}^1), \ldots, G_{p-1}(\zeta_{p-1}^N))$ random variables rather than being in sorted order.

The laws of the approximations $\hat{Z}_p^N$, $\eta_p^N(f)$ and $\hat{\eta}_p^N(f)$, however, are invariant to permutations of the indices of the ancestors $A_{p-1}^1,\ldots,A_{p-1}^N$ in the algorithm. Therefore, the sorting of the ancestor indices may be regarded as an implementation issue that does not affect the particle approximations themselves.