{"paper":{"title":"Orientation in Poisson Cluster Processes via Imaginary Bispectra","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A nonzero imaginary factorial bispectrum certifies orientation for stationary Poisson branching clusters when the reduced third cumulant is L1-integrable.","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Boris Baeumer, Conor Kresin, Ting Wang, Yifu Tang","submitted_at":"2026-05-13T04:59:10Z","abstract_excerpt":"We study what remains detectable about one-sided Poisson cluster processes after cluster orientation is erased. We construct matched reversible cluster nulls preserving intensity and the full Bartlett spectrum, showing that second-order structure alone need not identify temporal direction. For stationary Poisson branching clusters, we derive the Fourier--Stieltjes transform of the reduced third cumulant and show that, in the $L^1$ third-cumulant regime, a nonzero imaginary factorial bispectrum certifies orientation. We also give explicit orientation-erased nulls, reversible spectral matches fo"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For stationary Poisson branching clusters, in the L^1 third-cumulant regime, a nonzero imaginary factorial bispectrum certifies orientation.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The process belongs to the stationary Poisson branching cluster family and satisfies the L^1 integrability condition on the reduced third cumulant.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Nonzero imaginary factorial bispectrum certifies orientation in stationary Poisson branching clusters when the third cumulant is integrable.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A nonzero imaginary factorial bispectrum certifies orientation for stationary Poisson branching clusters when the reduced third cumulant is L1-integrable.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c6991f6635d817d87d87ded411e6d573b3b7ca7b5fe23591097e4e2ce48c438c"},"source":{"id":"2605.13004","kind":"arxiv","version":1},"verdict":{"id":"e38d7435-6e51-4968-9ec7-c3ad4fdc0527","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:34:31.716093Z","strongest_claim":"For stationary Poisson branching clusters, in the L^1 third-cumulant regime, a nonzero imaginary factorial bispectrum certifies orientation.","one_line_summary":"Nonzero imaginary factorial bispectrum certifies orientation in stationary Poisson branching clusters when the third cumulant is integrable.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The process belongs to the stationary Poisson branching cluster family and satisfies the L^1 integrability condition on the reduced third cumulant.","pith_extraction_headline":"A nonzero imaginary factorial bispectrum certifies orientation for stationary Poisson branching clusters when the reduced third cumulant is L1-integrable."},"references":{"count":25,"sample":[{"doi":"","year":2018,"title":"Achab, M., Bacry, E., Gaïffas, S., Mastromatteo, I. and Muzy, J.-F. (2018). Uncovering causality from multivariate Hawkes integrated cumulants.Journal of Machine Learning Research18(192), 1–28","work_id":"ab3b3a02-4381-4c91-b1ea-0f1ede64dd59","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"Bacry, E. and Muzy, J.-F. (2016). First- and second-order statistics characterization of Hawkes processes and non-parametric estimation.IEEE Transactions on Information Theory62, 2184–2202","work_id":"8ad767fd-97a4-4c10-b243-ba8b4d4ca559","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"Baddeley, A., Davies, T. M., Hazelton, M. L., Rakshit, S. and Turner, R. (2022). Fundamental problems in fitting spatial cluster process models.Spatial Statistics52, 100709","work_id":"c99d86d6-25f3-4cf8-a70e-4b88a56b99ea","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1987,"title":"Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987).Regular Variation. 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