ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

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A minimum-intensity coupling reduces a Poisson count correction

Example

For one channel with jump $ν = 1$ , compare a fine-step propensity $13.1$ and a coarse-step propensity $13$ over a half interval of length $h > 0$ . Couple the counts by a shared $Pois (13 h)$ variable and an independent fine-only residual $Pois (0.1 h)$ variable.

Facts & Assumptions

Given: Two Poisson marginal counts with parameters $13.1 h$ and $13 h$ , coupled by Shared-minimum coupling of two Poisson counts.

[F1]

The scalar Poisson count law: a Poisson count of parameter $θ \geq 0$ has mass $e^{- θ} θ^{k} / k!$ .

[F2]

Normalization, moments, and generating function of the Poisson count law: its mean and variance are both $θ$ .

[F3]

Shared-minimum coupling of two Poisson counts: shared Poisson counts plus independent rate-difference residuals have the required Poisson marginals and correction variance.

Verification

Given: The rates and step length in the Example.

1.1F1F3algebra

By [F3], the fine count is $P_{c} + P_{f}$ and the coarse count is $P_{c}$ , where the three count components have the independent laws in the Example. Their marginals have parameters $13.1 h$ and $13 h$ because independent Poisson counts add; this also follows by convolving the masses in [F1].

2.1F2step 1.1algebra∎

The coupled difference is exactly $P_{f}$ , so its variance is $0.1 h$ by [F2]. If the two counts were independent, their difference would have variance $13.1 h + 13 h = 26.1 h$ . Thus the shared minimum reduces the correction variance by a factor of $261$ in this one-step example.

Notes

This is a one-step count coupling example. It does not identify the fixed-step construction with T16's continuous-time reaction-path coupling.

Depends on

Used by

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Sources

David F. Anderson and Desmond J. Higham, Multilevel Monte Carlo for continuous time Markov chains, with applications in biochemical kinetics (standard reference, not scraped)