TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Conditional event second moments give a finite-particle SIR relative MSE bound

Statement

Assume AC. Use the full-multinomial-resampling event model of Full-resampling fixed-level SIR splitting with invariant mutations with $L \geq 1$ , $N \geq 1$ and positive prefix normalizers. Put $Z = Z_{L} = P (A_{L}) > 0$ , define $q_{L} = 1$ and $q_{p - 1} = M_{p} (G_{p} q_{p})$ for $1 \leq p \leq L$ , and let $η_{p}^{N}$ denote the weighted empirical state law after selection at level $p$ . Suppose deterministic finite constants $B_{0}, \dots, B_{L} \geq 1$ satisfy $P (q_{0}^{2}) \leq B_{0} Z^{2}$ and, almost surely on ${{\hat{Z}}_{p - 1} > 0}$ , $η_{p - 1}^{N} M_{p} [(G_{p} q_{p})^{2}] \leq B_{p} {η_{p - 1}^{N} M_{p} (G_{p} q_{p})}^{2}, 1 \leq p \leq L .$ Then $E [({\hat{Z}}_{L} - Z)^{2}] \leq Z^{2} {\prod_{p = 0}^{L} (1 + \frac{B_{p} - 1}{N}) - 1} .$ If $B_{0} = 1$ and $B_{p} \leq B$ for $1 \leq p \leq L$ , this is at most $Z^{2} [(1 + (B - 1) / N)^{L} - 1]$ . The conditions involve actual conditional event second moments under random empirical predictive laws. Bounds on squared conditional future means under the deterministic target laws alone do not imply them.

Facts & Assumptions

Given: The full-resampling event model, finite constants and conditional bounds in the Statement.

[A1]

The Axiom of Choice is inherited through the general SIR and conditional-expectation suppliers.

[F1]

Full-resampling fixed-level SIR splitting with invariant mutations and A bounded-potential SIR update has a uniform unnormalized variance bound give conditionally IID next proposals from $η_{p - 1}^{N} M_{p}$ and the identity ${\hat{Z}}_{p} η_{p}^{N} (f) = {\hat{Z}}_{p - 1} N^{- 1} \sum_{i} G_{p} (X_{p}^{i}) f (X_{p}^{i})$ , including zero weights.

[F2]

Full-resampling fixed-level SIR splitting with invariant mutations specifies initial law $P$ , initial potential $G_{0} = 1$ , probability transitions $M_{p}$ , and indicator potentials $G_{p}$ .

[F3]

Unbiasedness and exact variance of the IID Monte Carlo mean gives variance of a mean as common variance divided by $N$ . Moments, variance, and covariance on a probability space gives $E H^{2} = Var (H) + (E H)^{2}$ ; Conditional variance decomposition supplies conditional second-moment decomposition.

[F4]

Fixed-level full-resampling SIR splitting is unbiased gives $Z_{L} = P (A_{L}) = Z$ and $E {\hat{Z}}_{L} = Z$ .

Proof

1.1givenA1F1F2F3F4algebrainduction

Put $U_{p} = {\hat{Z}}_{p} η_{p}^{N} (q_{p})$ for $0 \leq p \leq L$ , and write $α_{p} = 1 + (B_{p} - 1) / N$ . Backward induction in the defining recursion gives $0 \leq q_{p} \leq 1$ ; expanding its finite ordered integrals gives $P (q_{0}) = Z_{L} = Z$ by [F2] and [F4]. The initial particles are IID from $P$ , ${\hat{Z}}_{0} = 1$ and $G_{0} = 1$ , so $U_{0} = N^{- 1} \sum_{i} q_{0} (X_{0}^{i})$ has mean $Z$ and second moment $Z^{2} + [P (q_{0}^{2}) - Z^{2}] / N \leq α_{0} Z^{2}$ . All $U_{p}$ lie in $[0, 1]$ by [F2], so all following conditional second moments exist.

1.2givenF1F2F3algebra

Conditional on the particle history $F_{p - 1}$ , put $z = {\hat{Z}}_{p - 1}$ , $a = η_{p - 1}^{N} M_{p} (G_{p} q_{p})$ and $b = η_{p - 1}^{N} M_{p} [(G_{p} q_{p})^{2}]$ . By [F1] and [F3], $E [U_{p}^{2} ∣ F_{p - 1}] = z^{2} [a^{2} + (b - a^{2}) / N]$ . On $z > 0$ , the hypothesis $b \leq B_{p} a^{2}$ bounds this by $α_{p} z^{2} a^{2} = α_{p} U_{p - 1}^{2}$ , using $q_{p - 1} = M_{p} (G_{p} q_{p})$ . If $a = 0$ this same calculation gives zero, without division by $a$ . If $z = 0$ , the zero-normalizer branch makes $U_{p} = U_{p - 1} = 0$ , so the inequality also holds without a hypothesis on the fallback empirical law.

2.1step 1.1step 1.2F3F4algebrainduction∎

Taking expectations in step 1.2 and iterating over $p = 1, \dots, L$ yields $E U_{L}^{2} \leq \prod_{p = 1}^{L} α_{p} E U_{0}^{2} \leq Z^{2} \prod_{p = 0}^{L} α_{p}$ . Since $q_{L} = 1$ , $U_{L} = {\hat{Z}}_{L}$ . Its mean is $Z$ by [F4], so subtracting $Z^{2}$ proves the claimed MSE bound. If $B_{0} = 1$ then $α_{0} = 1$ ; if every remaining $B_{p} \leq B$ , each remaining factor is at most $1 + (B - 1) / N$ , giving the specialization.

Source notes

The stated bound is proved here from the explicit conditional moment hypotheses, not attributed to the cited paper as an identical theorem. No new source reading is asserted. The cited nonasymptotic particle analysis is source context; its separate domination hypotheses are not inferred from these conditions.

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Sources

Cérou, Del Moral and Guyader, A nonasymptotic theorem for unnormalized Feynman–Kac particle models (standard reference, not scraped)