{ "schemaVersion": "1.6", "slug": "levy-qsd-exponential-moments-restart", "title": "Exact exponential moments of interval quasi-stationary laws for Lévy processes, and the saturation of idealised QSD restart", "shortTitle": "Restart from a quasi-stationary law: the exact constant and the real budget scale", "url": "https://evidencepress.org/releases/levy-qsd-exponential-moments-restart/", "oneLine": "A conjecture of Garcia, Bermolen, Jonckheere and Shneer on restart-based rare-event simulation holds up to a constant, but the constant is not one (0.850 in their own example), and idealised quasi-stationary restart saturates at budgets of order exp(ηL), far below the range stated in their restart theorem.", "abstract": "Restarting a simulation from a quasi-stationary distribution (QSD) is a simple way to look for a rare high level L of a process that drifts away from it, for example in rare-event simulation or in exploration for model-free simulation and reinforcement learning. Its efficiency is governed by an exponential moment of the QSD on the interval (0, L), and Garcia, Bermolen, Jonckheere and Shneer (Stochastic Systems, 2026) conjectured that this moment grows like exp((λ*−η)L), noting that this is exact for Brownian motion. For Lévy processes with a Gaussian component and jumps of either sign, this candidate proves that every exponential moment of the interval QSD has an exact limit after normalisation, with constants written through the ladder-height exponents of the Esscher-centred process; the core input is an exact L⁻² asymptotic for the two boundary layers of the centred QSD. The conjecture therefore holds on the logarithmic scale and up to constants, but its ratio-one reading fails: the constant is below one for every spectrally negative process with jumps, above one for every spectrally positive one, and 0.85014864899… in the source's own example (certified by interval arithmetic). For idealised QSD restart the success rate is asymptotic to R* exp(−ηL) with explicit R*, and the curve 1 − exp(−R* B exp(−ηL)) is accurate uniformly in the budget B in relative error, so the linear budget regime of the source's restart theorem is B = O(exp(ηL)), not B = o(exp(λ*L)). The moment theorem is extended to spectrally negative processes without a Gaussian part, and the first-order finite-L corrections are identified in the source model. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-10-21", "dateModified": "2026-10-21", "version": "1.0.0-candidate", "doi": "10.5281/zenodo.23545672", "doiUrl": "https://doi.org/10.5281/zenodo.23545672", "conceptDoi": null, "pdfUrl": "https://github.com/ipitchford/levy-qsd-exponential-moments-restart/releases/download/v1.0.0-candidate/levy-qsd-exponential-moments-restart-1.0.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23545672", "repoUrl": "https://github.com/ipitchford/levy-qsd-exponential-moments-restart", "releaseUrl": "https://github.com/ipitchford/levy-qsd-exponential-moments-restart/releases/tag/v1.0.0-candidate", "markdownUrl": "https://evidencepress.org/releases/levy-qsd-exponential-moments-restart/index.md", "bibtexUrl": "https://evidencepress.org/releases/levy-qsd-exponential-moments-restart/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/levy-qsd-exponential-moments-restart.mp3", "imageUrl": "https://evidencepress.org/assets/og/levy-qsd-exponential-moments-restart.png?v=a60ca78af8", "coverArtUrl": "https://evidencepress.org/assets/art/levy-qsd-exponential-moments-restart.svg?v=5da7786817", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/levy-qsd-exponential-moments-restart.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/levy-qsd-exponential-moments-restart.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematical candidate. Every theorem has a written proof (the finite-L corrections take a standard regularity property of the QSD density as given); the source-model constant is certified by interval arithmetic; numerical and exploratory results are labelled. Internal AI-assisted and adversarial review only. External specialist review, independent reproduction and formal verification have not been established." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23545672", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo. Provider asset parity was checked against frozen local files; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "verify_all.py regenerates every saved data output byte for byte. A separate clean replay of the full suite in an independent copy reproduced all outputs; the frozen archive's scripts and outputs were compared with that replay copy by hash, and only the changed checks were rerun before release." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "partial", "note": "Every constant is evaluated by two independent formulas at 70–85 digits, and the moment limits are checked against an exact four-mode finite-interval solution inside the producer workflow; this is implementation diversity, not external reproduction." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation. The enclosure of C* uses outward-rounded interval arithmetic, which is rigorous numerics, not formal verification." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal and adversarial review is not external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "Checks need Python 3.13 with NumPy, SciPy, SymPy, mpmath and Matplotlib; versions are recorded but not locked in a container." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical research", "AI-assisted internal review, verification re-review and adversarial review (including OpenAI gpt-5.6-sol)", "Claude Code: publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted and adversarial review, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the approved revised manuscript and its exact check outputs. Versioned GitHub and Zenodo research archives are checked against the frozen release assets on 21 October 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "quasi-stationary distribution", "Lévy processes", "rare-event simulation", "restart", "Esscher transform", "Wiener–Hopf factorisation", "ladder heights", "Fleming–Viot" ], "keyResults": [ "Exact moments (Theorem 2.2, proof): for a Lévy process with a Gaussian component, jumps of either sign and a Cramér root λ*, every exponential moment M_L(θ) of the QSD on (0, L) has an exact normalised limit: bounded for θ < η (recovering Kyprianou and Palmowski's half-line formula), of order L² at θ = η, and ∼ C(θ) exp((θ−η)L) for θ > η, with C(θ) given by ladder-height exponents of the Esscher-centred process.", "The engine (Theorem 2.3, proof): the centred QSD is a sine-shaped bulk with two thin boundary layers whose masses are exactly of order L⁻²; the left layer follows the ascending ladder renewal function, the right one the descending ladder. The conjecture's constant is C* = K_r(λ*−η)/K_ℓ(η) = G₊(λ*−η)/G₋(η).", "The ratio-one reading fails (Proposition 2.9, proof; certified enclosure): C* = 1 for Brownian motion, C* < 1 for every spectrally negative process with jumps, C* > 1 for every spectrally positive one, and C* = 0.85014864899… in the source's own example.", "Restart saturates at exp(ηL) (Theorem 2.6 and Corollary 2.7, proof): the success rate of idealised QSD restart is r_L ∼ R* exp(−ηL), the curve 1 − exp(−R* B exp(−ηL)) has relative error at most |a_L − 1| at every budget, and the linear regime of the source's restart theorem is B = O(exp(ηL)), not B = o(exp(λ*L)); an exact Brownian example refutes the wider range.", "Extensions: spectrally negative processes without a Gaussian part, C* = η²ψ″(η)/(2κ∞) ≤ 1 (Theorem 8.1, proof under a scale-function positivity condition); in the source model the killing rate sees an effective length L + δ₊ + δ₋ and both constants carry relative error π²(δ₋ − δ₊)/(2L) ≈ 0.497/L (Theorems 9.1–9.2, proof given standard QSD-density regularity)." ], "reviews": [], "evidencePackage": "Written proofs of every theorem; an outward-rounded interval-arithmetic enclosure of the source-model constant C* = 0.85014864899…; exact rational lattice checks and an exact Brownian counterexample to the stated budget range; 70–85-digit evaluations of all constants by two independent formulas; labelled high-precision finite-L numerics and scale-function computations; every saved output regenerated byte for byte by verify_all.py.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Garcia, Bermolen, Jonckheere and Shneer (2026), Efficiency of parallel and restart exploration strategies in model-free stochastic simulations. Stochastic Systems 16(3), 264–288 (arXiv:2503.03565).", "url": "https://doi.org/10.1287/stsy.2025.0108", "doi": "10.1287/stsy.2025.0108" }, { "citation": "Kyprianou and Palmowski (2006), Quasi-stationary distributions for Lévy processes. Bernoulli 12(4), 571–581.", "url": "https://doi.org/10.3150/bj/1155735927", "doi": "10.3150/bj/1155735927" }, { "citation": "Lambert (2000), Completely asymmetric Lévy processes confined in a finite interval. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 36(2), 251–274.", "url": "https://doi.org/10.1016/S0246-0203(00)00126-6", "doi": "10.1016/S0246-0203(00)00126-6" }, { "citation": "Kolb and Savov (2014), Exponential ergodicity of killed Lévy processes in a finite interval. Electronic Communications in Probability 19, paper 31.", "url": "https://doi.org/10.1214/ECP.v19-3006", "doi": "10.1214/ECP.v19-3006" }, { "citation": "Bertoin and Doney (1994), Cramér's estimate for Lévy processes. Statistics & Probability Letters 21(5), 363–365.", "url": "https://doi.org/10.1016/0167-7152(94)00032-8", "doi": "10.1016/0167-7152(94)00032-8" }, { "citation": "Villemonais (2014), General approximation method for the distribution of Markov processes conditioned not to be killed. ESAIM: Probability and Statistics 18, 441–467.", "url": "https://doi.org/10.1051/ps/2013045", "doi": "10.1051/ps/2013045" }, { "citation": "Cérou and Guyader (2007), Adaptive multilevel splitting for rare event analysis. Stochastic Analysis and Applications 25(2), 417–443.", "url": "https://doi.org/10.1080/07362990601139628", "doi": "10.1080/07362990601139628" }, { "citation": "Glasserman, Heidelberger, Shahabuddin and Zajic (1999), Multilevel splitting for estimating rare event probabilities. Operations Research 47(4), 585–600.", "url": "https://doi.org/10.1287/opre.47.4.585", "doi": "10.1287/opre.47.4.585" }, { "citation": "Ecoffet, Huizinga, Lehman, Stanley and Clune (2021), First return, then explore. Nature 590, 580–586.", "url": "https://doi.org/10.1038/s41586-020-03157-9", "doi": "10.1038/s41586-020-03157-9" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:levy-qsd-exponential-moments-restart", "attemptIds": [ "ep-attempt:levy-qsd-exponential-moments-restart-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "counterexample-proxy-first", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "other", "description": "A resolution of the Garcia–Bermolen–Jonckheere–Shneer conjecture on exponential moments of interval QSDs for Lévy processes with a Gaussian component (true up to an explicit constant, false in its ratio-one reading), with the exact restart constant and a corrected budget range for their restart theorem.", "scope": "Real Lévy processes with a Gaussian component, a finite cumulant near the Cramér root and continuous monitoring; spectrally negative processes also without a Gaussian part. Idealised restart with exact, free QSD draws; no claim for random walks, two-sided pure-jump processes or implementable Fleming–Viot restart." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "The conjecture, restart law and restart theorem are quoted from arXiv:2503.03565v3 (paragraph after Corollary 3.15, Lemma 6.2, Theorem 3.14 and Proof 6.4), with the journal numbering (Theorem 4, Corollary 2, Lemma 5) recorded in the manuscript; the source model's cumulant is reproduced exactly.", "remainingRisks": [ "Journal theorem numbers are mapped from the arXiv v3 text; the mathematics quoted is from v3.", "The finite-L corrections take a standard regularity property of the QSD density as given.", "The bounded literature check is not exhaustive priority certification." ] }, "humanJudgmentGates": [ "Publication is authorised; scientific endorsement is not inferred.", "External specialist review and independent reproduction remain open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "independentRerun", "specialistReview", "formalVerification", "priorityAssessment" ], "nextAction": "Have a specialist in Lévy fluctuation theory check the endpoint-mass theorem (Section 4, in particular the orientation of the ladder processes and the Wiener–Hopf cancellation) and the corrected budget range (Corollary 2.7).", "claimCeiling": "The Garcia et al. conjecture holds on the logarithmic scale and up to an explicit constant for Lévy processes with a Gaussian component; its ratio-one reading is false; idealised QSD restart saturates at budgets of order exp(ηL). 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