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Truncation and total-variation errors for stopped crossings

Statement

Assume AC. In the stopped-event setup of Finite-budget stopped-event kernel with a cemetery outcome, with $(E, E)$ standard Borel and initial probability law $μ$ , let $p = P_{μ} (τ_{T} < τ_{R})$ and let $p_{m}$ be the finite-budget normalizer of Finite-budget stopped-kernel measurability and event normalizer. Then $0 \leq p - p_{m} = P_{μ} (τ_{T} < τ_{R}, τ_{T} > m) \leq P_{μ} (τ_{T} \land τ_{R} > m) .$ Thus the finite-budget stopped normalizer estimates the truncated event; a rate for its bias requires a tail bound for the stopping time. Separately, assume AC and let both finite-horizon Feynman–Kac models have the same finite horizon $H \in N_{0}$ , the same standard Borel state spaces $(X_{j}, X_{j})$ for $0 \leq j \leq H$ , the same initial law $μ_{0}$ , and the same potentials $G_{j} : X_{j} \to [0, 1]$ for $0 \leq j \leq H$ . Let their transition kernels be $M_{j}, {\tilde{M}}_{j} : X_{j - 1} ⇝ X_{j}$ for $1 \leq j \leq H$ , and suppose

\sup_{x} ∥ M_{j} (x, \cdot) - {\tilde{M}}_{j} (x, \cdot) ∥_{T V} \leq ϵ_{j}, ϵ_{j} \geq 0,

where $∥ ν - ν^{'} ∥_{T V} = \sup_{B} ∣ ν (B) - ν^{'} (B) ∣$ . If $Z_{H}^{M}$ and $Z_{H}^{\tilde{M}}$ are the normalizers in the common Feynman–Kac setup of Feynman–Kac path targets and normalizers, then $∣ Z_{H}^{M} - Z_{H}^{\tilde{M}} ∣ \leq \sum_{j = 1}^{H} ϵ_{j}$ . If a grid-detected crossing event $A_{h}$ is contained in a continuous-time target event $A$ , its probability bias is exactly $P (A ∖ A_{h})$ ; no grid-error rate follows without a separate path regularity or crossing-probability bound.

Facts & Assumptions

Given: AC, the standard Borel stopped-event model and its finite-budget probability, and two common finite-horizon Feynman–Kac models with identical initial law, state spaces, and potentials, differing only in transition kernels by the stated total-variation bounds.

[F1]

Finite-budget stopped-event kernel with a cemetery outcome: $\partial$ is the budget-censored outcome and $κ_{m}$ records success only when the target is hit before return by time $m$ .

[F2]

Measure kernel and probability kernel: A probability kernel has probability-measure rows and measurable evaluations.

[F3]

Probability measures and probability spaces: Probability measures have total mass one and are countably additive.

[F4]

Finite-budget stopped-kernel measurability and event normalizer: The finite-budget hit probability is $p_{m} = P_{μ} (τ_{T} < τ_{R}, τ_{T} \leq m)$ .

[F5]

Measurability of integration against a kernel: Integrating a bounded measurable function against a kernel gives a measurable function of the source state.

[F6]

Feynman–Kac path targets and normalizers: AC and a finite sequence of standard Borel state spaces, one initial law, and the ordered transition kernels and potentials define each finite-horizon Feynman–Kac normalizer.

Proof

1.1F1F3F4given

By [F4], $p_{m} = P_{μ} (τ_{T} < τ_{R}, τ_{T} \leq m)$ . The events increase to ${τ_{T} < τ_{R}}$ , and the omitted part at budget $m$ is ${τ_{T} < τ_{R}, τ_{T} > m} \subseteq {τ_{T} \land τ_{R} > m}$ . Countable additivity [F3] gives the displayed difference and tail bound.

1.2F2F3F5F6givenalgebra

Replace the transition kernels one at a time and telescope the difference of the two normalizers. For the term replacing $M_{j}$ , the earlier kernels and potentials define a finite measure of mass at most one, because the initial law is a probability and every potential lies in $[0, 1]$ . The remaining kernel-potential composition is a measurable function $g_{j}$ with $0 \leq g_{j} \leq 1$ , measurable by [F5] and kernel composition. By the layer-cake identity $g_{j} (y) = \int_{0}^{1} 1_{{g_{j} (y) > t}} d t$ , the total-variation definition bounds $∣ M_{j} g_{j} (x) - {\tilde{M}}_{j} g_{j} (x) ∣ \leq ϵ_{j}$ for every $x$ . Integrating against that finite measure bounds the telescoping term by $ϵ_{j}$ ; summing yields $\sum_{j} ϵ_{j}$ .

2.1F3givenalgebra∎

If $A_{h} \subseteq A$ , then $P (A) - P (A_{h}) = P (A ∖ A_{h})$ by finite additivity on disjoint measurable sets. A bound on this value requires an additional estimate for the missed-crossing event; continuity of a path alone does not supply a uniform grid-rate without quantitative regularity.

Depends on

Used by

A finite grid can miss a continuous-time crossing Example

Dependency tree · two levels

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Sources

Adam M. Johansen, Pierre Del Moral, and Arnaud Doucet, Sequential Monte Carlo Samplers for Rare Events, `3 (standard reference, not scraped)
Frédéric Cérou, Pierre Del Moral, and Arnaud Guyader, A nonasymptotic theorem for unnormalized Feynman–Kac particle models, Theorem 1.5 and `5 (standard reference, not scraped)