TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Fixed-horizon SMC central limit theorem with multinomial resampling

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 the SIR procedure with multinomial resampling at each time. 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 $\sqrt{N} {ω_{t}^{N} (ϕ) - π_{t} (ϕ)} \Rightarrow N_{d} (0, V_{t} (ϕ)), \sqrt{N} {{\overset{ˇ}{η}}_{t}^{N} (ϕ) - π_{t} (ϕ)} \Rightarrow N_{d} (0, {\hat{V}}_{t}^{m u l t i} (ϕ)),$ where ${\hat{V}}_{t}^{m u l t i} (ϕ) = V_{t} (ϕ) + {Cov}_{π_{t}} (ϕ)$ . The horizon is fixed as $N \to \infty$ ; the theorem asserts no time-uniform finite-particle accuracy.

Facts & Assumptions

Given: A fixed finite SIR horizon with multinomial selection, the recursive target-moment class $Ψ_{t}^{(d)}$ , and the source normalizer condition $1 \in Φ_{s}^{(1)}$ at every time through $t$ .

[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 each centered forward image in $Ψ_{t - 1}$ ; the source requires $1 \in Φ_{s}$ ; $V_{t}$ and ${\hat{V}}_{t}^{m u l t i}$ are the stated covariance recursions.

[F2]

Conditional mutation and weighted correction CLT: Given the preceding selected-population CLT, the mutation and correction steps give the weighted-estimator CLT with covariance $V_{s}$ ; the initial weighted proposal has the ratio CLT under $Ψ_{0}$ .

[F3]

Conditional multinomial resampling central limit theorem: Given the weighted-estimator CLT, a finite target second moment, and the empirical-law conclusions of [F5], multinomial selection adds ${Cov}_{π_{s}} (ϕ)$ 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 multinomial 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 same 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 target moment in $Ψ_{0}$ and the weighted CLT meet [F3], which gives the post-resampling CLT with variance $V_{0} (h) + {Var}_{π_{0}} (h)$ .

2.2F1F2F3step 1.1step 1.2

The target moment $h \in Ψ_{s}$ , the weighted CLT just obtained, and the empirical laws from step 1.1 satisfy [F3]. Multinomial resampling therefore gives the selected CLT with covariance $V_{s} (h) + {Var}_{π_{s}} (h) = {\hat{V}}_{s}^{m u l t i} (h)$ . This proves the induction assertion for every $h \in Ψ_{s}$ .

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 $a^{⊤} {\hat{V}}_{t}^{m u l t i} (ϕ) a$ . Since $a$ was arbitrary, [F4] gives the two vector limits. No bound uniform over the horizon follows from this fixed- $t$ argument.

Depends on

Used by

Fixed-horizon asymptotic variance can grow without mixing Example

Dependency tree · two levels

35 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Nicolas Chopin, Central Limit Theorem for Sequential Monte Carlo Methods and its Application to Bayesian Inference, Theorem 1 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)