{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T09:12:34Z","timestamp":1775466754143,"version":"3.50.1"},"reference-count":38,"publisher":"Institute of Mathematical Statistics","issue":"none","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Probab."],"published-print":{"date-parts":[[2024,1,1]]},"DOI":"10.1214\/24-ejp1231","type":"journal-article","created":{"date-parts":[[2024,11,8]],"date-time":"2024-11-08T13:56:28Z","timestamp":1731074188000},"source":"Crossref","is-referenced-by-count":1,"title":["Convergence to decorated L\u00e9vy processes in non-Skorohod topologies for dynamical systems"],"prefix":"10.1214","volume":"29","author":[{"given":"Ana Cristina","family":"Moreira Freitas","sequence":"first","affiliation":[{"name":"Centro de Matem\u00e1tica & Faculdade de Economia da Universidade do Porta, Portugal"}]},{"given":"Jorge","family":"Milhazes Freitas","sequence":"additional","affiliation":[{"name":"Centro de Matem\u00e1tica & Faculdade de Ci\u00eancias da Universidade do Porto, Portugal"}]},{"given":"Ian","family":"Melbourne","sequence":"additional","affiliation":[{"name":"Mathematics Institute, University of Warwick, UK"}]},{"given":"Mike","family":"Todd","sequence":"additional","affiliation":[{"name":"Mathematical Institute, University of St Andrews, Scotland"}]}],"member":"108","reference":[{"key":"21","doi-asserted-by":"publisher","unstructured":"D. Kelly and I. Melbourne. Smooth approximation of stochastic differential equations. <i>Ann. Probab.<\/i> <b>44<\/b> (2016) 479\u2013520.","DOI":"10.1214\/14-AOP979"},{"key":"23","doi-asserted-by":"publisher","unstructured":"C. Liverani, B. Saussol and S. Vaienti. A probabilistic approach to intermittency. <i>Ergodic Theory Dynam. Systems<\/i> <b>19<\/b> (1999) 671\u2013685.","DOI":"10.1017\/S0143385799133856"},{"key":"30","doi-asserted-by":"crossref","unstructured":"Y. Pomeau and P. Manneville. Intermittent transition to turbulence in dissipative dynamical systems. <i>Comm. Math. Phys.<\/i> <b>74<\/b> (1980) 189\u2013197.","DOI":"10.1007\/BF01197757"},{"key":"37","doi-asserted-by":"publisher","unstructured":"L.-S. Young. Statistical properties of dynamical systems with some hyperbolicity. <i>Ann. of Math.<\/i> <b>147<\/b> (1998) 585\u2013650.","DOI":"10.2307\/120960"},{"key":"38","doi-asserted-by":"publisher","unstructured":"R. Zweim\u00fcller. Mixing limit theorems for ergodic transformations. <i>J. Theoret. Probab.<\/i> <b>20<\/b> (2007) 1059\u20131071.","DOI":"10.1007\/s10959-007-0085-y"},{"key":"9","doi-asserted-by":"publisher","unstructured":"I. Chevyrev, P. K. Friz, A. Korepanov and I. Melbourne. Superdiffusive limits for deterministic fast-slow dynamical systems. <i>Probab. Theory Related Fields<\/i> <b>178<\/b> (2020) 735\u2013770.","DOI":"10.1007\/s00440-020-00988-5"},{"key":"14","unstructured":"S. Gou\u00ebzel. Stable laws for the doubling map. Unpublished notes."},{"key":"25","doi-asserted-by":"publisher","unstructured":"I. Melbourne and A. T\u00f6r\u00f6k. Convergence of moments for axiom a and nonuniformly hyperbolic flows. <i>Ergodic Theory Dynam. Systems<\/i> <b>32<\/b> (2012) 1091\u20131100.","DOI":"10.1017\/S0143385711000174"},{"key":"1","doi-asserted-by":"crossref","unstructured":"F. Avram and M. S. Taqqu. Weak convergence of sums of moving averages in the <i>\u03b1<\/i>-stable domain of attraction. <i>Ann. Probab.<\/i> <b>20<\/b> (1992) 483\u2013503.","DOI":"10.1214\/aop\/1176989938"},{"key":"2","doi-asserted-by":"crossref","unstructured":"P. B\u00e1lint, N. Chernov and D. Dolgopyat. Limit theorems for dispersing billiards with cusps. <i>Comm. Math. Phys.<\/i> <b>308<\/b> (2011) 479\u2013510.","DOI":"10.1007\/s00220-011-1342-6"},{"key":"3","doi-asserted-by":"crossref","unstructured":"F. Bartumeus and S. A. Levin. Fractal reorientation clocks: Linking animal behavior to statistical patterns of search. <i>Proc. Natl. Acad. Sci. USA<\/i> <b>105<\/b> (2008) 19072\u201319077.","DOI":"10.1073\/pnas.0801926105"},{"key":"4","doi-asserted-by":"crossref","unstructured":"B. Basrak and D. Krizmani\u0107. A limit theorem for moving averages in the <i>\u03b1<\/i>-stable domain of attraction. <i>Stochastic Process. Appl.<\/i> <b>124<\/b> (2014) 1070\u20131083.","DOI":"10.1016\/j.spa.2013.10.006"},{"key":"5","doi-asserted-by":"crossref","unstructured":"B. Basrak, D. Krizmani\u0107 and J. Segers. A functional limit theorem for dependent sequences with infinite variance stable limits. <i>Ann. Probab.<\/i> <b>40<\/b> (2012) 2008\u20132033.","DOI":"10.1214\/11-AOP669"},{"key":"6","doi-asserted-by":"crossref","unstructured":"L. A. Bunimovich and Y. G. Sina\u012d. Statistical properties of Lorentz gas with periodic configuration of scatterers. <i>Comm. Math. Phys.<\/i> <b>78<\/b> (1980\/81) 479\u2013497.","DOI":"10.1007\/BF02046760"},{"key":"7","doi-asserted-by":"crossref","unstructured":"L. A. Bunimovich, Y. G. Sina\u012d and N. I. Chernov. Statistical properties of two-dimensional hyperbolic billiards. <i>Uspekhi Mat. Nauk<\/i> <b>46<\/b> (1991) 43\u201392.","DOI":"10.1070\/RM1991v046n04ABEH002827"},{"key":"8","doi-asserted-by":"crossref","unstructured":"N. Chernov and R. Markarian. <i>Chaotic billiards<\/i>. Mathematical Surveys and Monographs <b>127<\/b>, American Mathematical Society, Providence, RI, 2006.","DOI":"10.1090\/surv\/127"},{"key":"10","unstructured":"I. Chevyrev, A. Korepanov and I. Melbourne. Superdiffusive limits beyond the Marcus regime for deterministic fast-slow dynamical systems. Preprint (<a href=\"https:\/\/arxiv.org\/abs\/2312.15734\">https:\/\/arxiv.org\/abs\/2312.15734<\/a>). To appear in <i>Comm. Amer. Math. Soc<\/i>."},{"key":"11","doi-asserted-by":"crossref","unstructured":"A. C. M. Freitas, J. M. Freitas and M. Magalh\u00e3es. Complete convergence and records for dynamically generated stochastic processes. <i>Trans. Amer. Math. Soc.<\/i> <b>373<\/b> (2020) 435\u2013478.","DOI":"10.1090\/tran\/7922"},{"key":"12","unstructured":"A. C. M. Freitas, J. M. Freitas and M. Todd. Enriched functional limit theorems for dynamical systems. Preprint (<a href=\"https:\/\/arxiv.org\/abs\/2011.10153\">https:\/\/arxiv.org\/abs\/2011.10153<\/a>)."},{"key":"13","doi-asserted-by":"crossref","unstructured":"P. Gaspard and X.-J. Wang. Sporadicity: Between periodic and chaotic dynamical behaviors. <i>Proc. Natl. Acad. Sci. USA<\/i> <b>85<\/b> (1988) 4591\u20134595.","DOI":"10.1073\/pnas.85.13.4591"},{"key":"15","doi-asserted-by":"crossref","unstructured":"S. Gou\u00ebzel. Central limit theorem and stable laws for intermittent maps. <i>Probab. Theory Relat. Fields<\/i> <b>128<\/b> (2004) 82\u2013122.","DOI":"10.1007\/s00440-003-0300-4"},{"key":"16","doi-asserted-by":"crossref","unstructured":"S. Gou\u00ebzel. Statistical properties of a skew product with a curve of neutral points. <i>Ergodic Theory Dynam. Systems<\/i> <b>27<\/b> (2007) 123\u2013151.","DOI":"10.1017\/S0143385706000617"},{"key":"17","unstructured":"A. Jakubowski. The Skorokhod Space in functional convergence: a short introduction. <i>International conference: Skorokhod Space. 50 years on, 17-23 June 2007, Kyiv, Ukraine, Part I, s. 11-18.<\/i>, 2007."},{"key":"18","doi-asserted-by":"crossref","unstructured":"P. Jung, I. Melbourne, F. P\u00e8ne, P. Varandas and H.-K. Zhang. Necessary and sufficient condition for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\" id=\"eq_1\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math>-convergence to a L\u00e9vy process for billiards with cusps at flat points. <i>Stoch. Dyn.<\/i> <b>21<\/b> (2021) 2150024, 8 pages.","DOI":"10.1155\/2021\/5542787"},{"key":"19","doi-asserted-by":"crossref","unstructured":"P. Jung, F. P\u00e8ne and H.-K. Zhang. Convergence to <i>\u03b1<\/i>-stable L\u00e9vy motion for chaotic billiards with several cusps at flat points. <i>Nonlinearity<\/i> <b>33<\/b> (2020) 807\u2013839.","DOI":"10.1088\/1361-6544\/ab5148"},{"key":"20","doi-asserted-by":"crossref","unstructured":"P. Jung and H.-K. Zhang. Stable laws for chaotic billiards with cusps at flat points. <i>Annales Henri Poincar\u00e9<\/i> <b>19<\/b> (2018) 3815\u20133853.","DOI":"10.1007\/s00023-018-0726-y"},{"key":"22","doi-asserted-by":"crossref","unstructured":"R. Klages, S. Gallegos, J. Solanp\u00e4\u00e4, M. Sarvilahti and E R\u00e4s\u00e4nen. Normal and anomalous diffusion in soft Lorentz gases. <i>Phys. Rev. Lett.<\/i> <b>122<\/b> (2019) 064102.","DOI":"10.1103\/PhysRevLett.122.064102"},{"key":"24","unstructured":"H Lorentz. The motion of electrons in metallic bodies. <i>Proc. Amst. Acad.<\/i> <b>7<\/b> (1905) 438\u2013453."},{"key":"26","doi-asserted-by":"crossref","unstructured":"I. Melbourne and P. Varandas. Convergence to a L\u00e9vy process in the Skorohod <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\" id=\"eq_2\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\" id=\"eq_3\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> topologies for nonuniformly hyperbolic systems, including billiards with cusps. <i>Comm. Math. Phys.<\/i> <b>375<\/b> (2020) 653\u2013678.","DOI":"10.1155\/2020\/7126045"},{"key":"27","doi-asserted-by":"crossref","unstructured":"I. Melbourne and R. Zweim\u00fcller. Weak convergence to stable L\u00e9vy processes for nonuniformly hyperbolic dynamical systems. <i>Ann Inst. H. Poincar\u00e9 (B) Probab. Statist.<\/i> <b>51<\/b> (2015) 545\u2013556.","DOI":"10.1214\/13-AIHP586"},{"key":"28","doi-asserted-by":"crossref","unstructured":"R. Metzler, J-H. Jeon, A. G. Cherstvy and E. Barkai. Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. <i>Phys. Chem. Chem. Phys.<\/i> <b>16<\/b> (2014) 24128\u201324164.","DOI":"10.1039\/C4CP03465A"},{"key":"29","doi-asserted-by":"crossref","unstructured":"B. Podobnik, A. Valentinc\u012dc\u02d8, D. Horvati\u0107 and H. E. Stanley. Asymmetric L\u00e9vy flight in financial ratios. <i>Proc. Natl. Acad. Sci. USA<\/i> <b>108<\/b> (2011) 17883\u201317888.","DOI":"10.1073\/pnas.1113330108"},{"key":"31","unstructured":"G. Samorodnitsky and M. S. Taqqu. <i>Stable non-Gaussian random processes<\/i>, Stochastic Modeling. Chapman &amp; Hall, New York, 1994. Stochastic models with infinite variance."},{"key":"32","unstructured":"Y. G. Sina\u012d. Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. <i>Uspehi Mat. Nauk<\/i> <b>25<\/b> (1970) 141\u2013192."},{"key":"33","unstructured":"A. V. Skorohod. Limit theorems for stochastic processes. <i>Teor. Veroyatnost. i Primenen.<\/i> <b>1<\/b> (1956) 289\u2013319."},{"key":"34","doi-asserted-by":"crossref","unstructured":"T. H. Solomon, E. R. Weeks and H. L. Swinney. Chaotic advection in a two-dimensional flow: L\u00e9vy flights and anomalous diffusion. <i>Phys. D<\/i> <b>76<\/b> (1994) 70\u201384.","DOI":"10.1016\/0167-2789(94)90251-8"},{"key":"35","doi-asserted-by":"crossref","unstructured":"M. Tyran-Kami\u0144ska. Weak convergence to L\u00e9vy stable processes in dynamical systems. <i>Stoch. Dyn.<\/i> <b>10<\/b> (2010) 263\u2013289.","DOI":"10.1142\/S0219493710002942"},{"key":"36","doi-asserted-by":"crossref","unstructured":"W. Whitt. <i>Stochastic-process limits<\/i>. Springer Series in Operations Research, Springer-Verlag, New York, 2002. An introduction to stochastic-process limits and their application to queues.","DOI":"10.1007\/b97479"}],"container-title":["Electronic Journal of Probability"],"original-title":[],"deposited":{"date-parts":[[2024,11,8]],"date-time":"2024-11-08T13:56:43Z","timestamp":1731074203000},"score":1,"resource":{"primary":{"URL":"https:\/\/projecteuclid.org\/journals\/electronic-journal-of-probability\/volume-29\/issue-none\/Convergence-to-decorated-L%c3%a9vy-processes-in-non-Skorohod-topologies-for\/10.1214\/24-EJP1231.full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,1]]},"references-count":38,"journal-issue":{"issue":"none","published-online":{"date-parts":[[2024,1,1]]}},"URL":"https:\/\/doi.org\/10.1214\/24-ejp1231","relation":{},"ISSN":["1083-6489"],"issn-type":[{"value":"1083-6489","type":"print"}],"subject":[],"published":{"date-parts":[[2024,1,1]]}}}