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Fixed-horizon SMC central limit theorem with residual resampling away from integer weights

Statement

Fix a finite horizon $t$ and use the target/proposal flow and classes $Φ_{s}^{(d)}, Ψ_{s}^{(d)}$ of Finite-horizon SMC CLT classes and variance recursions, with $1 \in Φ_{s}^{(1)}$ for every $0 \leq s \leq t$ . Run SIR with residual resampling at each time. At every $0 \leq s \leq t$ , assume either $υ_{s} = 1$ ${\tilde{π}}_{s}$ -almost everywhere, or ${\tilde{π}}_{s} (υ_{s} \in {1, 2, \dots}) = 0$ . For $ϕ \in Ψ_{t}^{(d)}$ , let $ω_{t}^{N}$ be the corrected weighted empirical law and ${\overset{ˇ}{η}}_{t}^{N}$ the post-resampling empirical law. Then $\sqrt{N} {ω_{t}^{N} (ϕ) - π_{t} (ϕ)} \Rightarrow N_{d} (0, V_{t} (ϕ)), \sqrt{N} {{\overset{ˇ}{η}}_{t}^{N} (ϕ) - π_{t} (ϕ)} \Rightarrow N_{d} (0, {\hat{V}}_{t}^{r e s} (ϕ)),$ where ${\hat{V}}_{s}^{r e s} = V_{s} + R_{s}$ is defined by Finite-horizon SMC CLT classes and variance recursions in the stated branches. No conclusion is asserted for a nonconstant weight law with positive mass at a positive integer. The horizon is fixed as $N \to \infty$ .

Facts & Assumptions

Given: A fixed finite SIR horizon with residual selection, the recursive target-moment class $Ψ_{t}^{(d)}$ , the source condition $1 \in Φ_{s}^{(1)}$ at every time, and the no-positive-integer-atom or constant-weight condition at every time.

[A1]

The Axiom of Choice: AC is inherited through the finite-horizon path-law and particle-law construction.

[F1]

Finite-horizon SMC CLT classes and variance recursions: $Ψ_{t}$ includes the terminal target moment and recursively places centered forward images in $Ψ_{t - 1}$ ; the source condition is $1 \in Φ_{s}$ ; the displayed residual covariance is assigned only in the stated branches.

[F2]

Conditional mutation and weighted correction CLT: Given the preceding selected-population CLT, mutation and correction give the weighted-estimator CLT with covariance $V_{s}$ , with the initial ratio CLT at $s = 0$ .

[F3]

Conditional residual resampling central limit theorem away from integer weights: Under proposal empirical-law convergence and the stated residual-weight condition, residual selection adds $R_{s}$ to the weighted-estimator variance and gives the selected-population CLT.

[F4]

Cramer wold device: Convergence of all scalar linear projections to the corresponding centered Gaussian projections implies vector weak convergence to the multivariate Gaussian with that covariance matrix.

[F5]

Finite-horizon SMC empirical laws converge for integrable tests: In the stated finite-horizon SIR flow with multinomial or residual selection, proposal empirical averages converge in probability for every proposal-integrable test, and corrected and selected empirical averages do so for every target-integrable test, at each fixed time.

Proof

Given: The hypotheses in the Statement.

1.1A1F1F5givenalgebra

The finite-horizon SIR flow and residual selection satisfy [F5], so all its integrable-test empirical-law conclusions hold at every $s \leq t$ , as required by [F2] and [F3]. Fix $a \in R^{d}$ and put $h = a^{⊤} ϕ$ . The bound $∣ a^{⊤} x ∣ \leq ∥ a ∥ ∥ x ∥$ and linearity of each $E_{s}$ show from [F1] that $h \in Ψ_{t}^{(1)}$ . We prove by induction on $s \leq t$ the scalar assertion for every test in $Ψ_{s}^{(1)}$ ; at $s = t$ this includes $h$ .

1.2F1F2giveninduction

Let $1 \leq s \leq t$ , assume the assertion holds for every test in $Ψ_{s - 1}$ , and take arbitrary $h \in Ψ_{s}$ . By [F1], $E_{s} (h - π_{s} h) \in Ψ_{s - 1}$ , so the induction hypothesis gives the selected CLT needed by [F2] for the centered weighted test. The source condition $1 \in Φ_{s}$ makes the correction ratio well defined in the stated class. Thus [F2] yields the weighted CLT at time $s$ with covariance $V_{s} (h)$ .

2.1F1F2F3step 1.1

At time zero, let $h \in Ψ_{0}$ . By [F2], the initial weighted ratio has CLT variance $V_{0} (h)$ . The assumed residual-weight branch either has constant weight one, in which case selection is deterministic and $R_{0} (h) = 0$ , or has no positive-integer atom; in the latter case the target moment and weighted CLT meet [F3], which gives the selected CLT with its stated $R_{0} (h)$ .

2.2F1F2F3step 1.1step 1.2

The target moment $h \in Ψ_{s}$ , the weighted CLT just obtained, the empirical laws from step 1.1, and the residual-weight condition satisfy [F3]. Residual selection gives the selected CLT with covariance $V_{s} (h) + R_{s} (h) = {\hat{V}}_{s}^{r e s} (h)$ . This proves the induction assertion for every $h \in Ψ_{s}$ while retaining the stated weight condition at each time.

3.1F1F4step 1.1step 2.1step 1.2step 2.2∎

The base step 2.1 and induction steps 1.2, 2.2 prove both scalar limits at time $t$ for every $h \in Ψ_{t}$ . For each $a \in R^{d}$ , $a^{⊤} ϕ \in Ψ_{t}$ by step 1.1 and has covariance projections $a^{⊤} V_{t} (ϕ) a$ and $a^{⊤} {\hat{V}}_{t}^{r e s} (ϕ) a$ . Since $a$ was arbitrary, [F4] yields the two vector limits. The integer-boundary branch is excluded by hypothesis and receives no conclusion here.

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Sources

Nicolas Chopin, Central Limit Theorem for Sequential Monte Carlo Methods and its Application to Bayesian Inference, Theorem 2 and Appendix A.1 (standard reference, not scraped)
Hans R. Künsch, Recursive Monte Carlo Filters: Algorithms and Theoretical Analysis, §4, Theorem 4 (standard reference, not scraped)
Arnaud Doucet and Adam M. Johansen, A Tutorial on Particle Filtering and Smoothing: Fifteen Years Later, §3.6 (standard reference, not scraped)