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Clipping an understated thinning envelope changes the waiting-time law

Statement

Assume Countable Choice and Dependent Choice. Consider the one-state system $E = {0}$ with one zero-increment channel $ν = 0$ and constant propensity $a (t, 0) = 2$ . Its first marked reaction time has survival $e^{- 2 t}$ . If an implementation incorrectly uses the understated envelope $Λ = 1$ and clips the acceptance probability $a / Λ = 2$ to $1$ , it accepts every rate-one candidate, so its first marked reaction time has survival $e^{- t}$ . These laws differ for every $t > 0$ . The specified algorithm requires a valid envelope and does not clip: a detected violation returns invalid-envelope, and the exact-thinning theorem asserts no law under a failed envelope.

Facts & Assumptions

Given: Countable Choice, Dependent Choice, and the one-state time-dependent propensity system above and the candidate-thinning algorithm.

[F1]

A zero increment is an allowed marked reaction that leaves the state unchanged, while its propensity still contributes to the total event rate. (Finite-channel reaction propensity systems and their generator)

[F2]

The thinning algorithm requires a supplied envelope certificate, reports a detected violation as invalid-envelope, and does not clip propensities. (Dominating thinning for bounded time-varying propensities)

[F3]

Under Countable Choice and Dependent Choice, exact thinning conclusions require the certified envelope; if it fails, the theorem asserts no path law. (Dominating thinning has the prescribed reaction law)

[F4]

Under Countable Choice and Dependent Choice, for a finite-state reaction system with a specified initial state and unlimited event budget, the Gillespie direct method uses an exponential holding time with parameter equal to the total propensity. (Finite-state Gillespie and next-reaction methods have the specified CTMC law)

Proof

technique · direct

1.1F1F4givenalgebra

The target system has constant total rate $2$ ; [F4] therefore gives its first marked-event survival $e^{- 2 t}$ . This is an event even though $ν = 0$ , by [F1].

2.1F2step 1.1algebra

A rate-one candidate process has no candidate by time $t$ with probability $e^{- t}$ . The clipped rule accepts every candidate, hence its first accepted-event survival is $e^{- t}$ ; for $t > 0$ , $e^{- t} \neq e^{- 2 t}$ . Thus clipping changes the event law.

3.1F2F3step 2.1given∎

Here $a = 2 > Λ = 1$ , so the required certificate is false. The specified procedure therefore lies outside the theorem's hypothesis and, whenever the violation is evaluated at a candidate, returns invalid-envelope; [F2] expressly forbids continuing by clipping.

Depends on

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Sources

David F. Anderson and Thomas G. Kurtz, Continuous Time Markov Chain Models for Chemical Reaction Networks (standard reference, not scraped)