{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:55:25Z","timestamp":1777449325957,"version":"3.51.4"},"reference-count":23,"publisher":"SAGE Publications","issue":"1-2","license":[{"start":{"date-parts":[[2016,3,9]],"date-time":"2016-03-09T00:00:00Z","timestamp":1457481600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2016,3,9]]},"abstract":"<jats:p>The law of the iterated logarithm for some Markov operators, which converge exponentially to the invariant measure, is established. The operators correspond to iterated function systems which, for example, may be used to generalize the cell cycle model examined by Lasota and Mackey [ J. Math. Biol. 38 (1999), 241\u2013261].<\/jats:p>","DOI":"10.3233\/asy-151344","type":"journal-article","created":{"date-parts":[[2016,3,11]],"date-time":"2016-03-11T09:17:02Z","timestamp":1457687822000},"page":"91-112","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":4,"title":["Law of the iterated logarithm for some Markov operators"],"prefix":"10.1177","volume":"97","author":[{"given":"Sander","family":"Hille","sequence":"first","affiliation":[{"name":"Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, Netherlands. E-mail:\u00a0"}]},{"given":"Katarzyna","family":"Horbacz","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland. E-mail:\u00a0"}]},{"given":"Tomasz","family":"Szarek","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, University of Gda\u0144sk, Wita Stwosza 57, 80-952 Gda\u0144sk, Poland. E-mail:\u00a0"}]},{"given":"Hanna","family":"Wojew\u00f3dka","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, Institute of Theoretical Physics and Astrophysics, University of Gda\u0144sk, Wita Stwosza 57, 80-952 Gda\u0144sk, Poland"},{"name":"National Quantum Information Center of Gda\u0144sk, ul. W\u0142. Andersa 27, 81-824 Sopot, Poland. E-mail:\u00a0"}]}],"member":"179","published-online":{"date-parts":[[2016,3,9]]},"reference":[{"key":"ref001","unstructured":"[1]P.Billingsley, Convergence of Probability Measures, Wiley, New York, 1968."},{"key":"ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-34514-5"},{"key":"ref003","doi-asserted-by":"publisher","DOI":"10.4064\/sm212-1-3"},{"key":"ref004","doi-asserted-by":"publisher","DOI":"10.24033\/asens.1013"},{"key":"ref005","doi-asserted-by":"publisher","DOI":"10.1007\/s004400200216"},{"key":"ref006","unstructured":"[6]P.Hall and C.C.Heyde, Martingale Limit Theory and Its Application, Academic Press, New York, 1980."},{"key":"ref007","unstructured":"[7]P.R.Halmos, Measure Theory, Springer-Verlag, New York, 1974."},{"key":"ref008","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176996937"},{"key":"ref009","unstructured":"[9]S.C.Hille, K.Horbacz and T.Szarek, Existence of a unique invariant measure for a class of equicontinuous Markov operators with application to a stochastic model for an autoregulated gene,\n                      Ann. Math. Blaise Pascal\n                      , to appear."},{"key":"ref010","unstructured":"[10]S.C.Hille, K.Horbacz, T.Szarek and H.Wojew\u00f3dka, Limit theorems for some Markov operators, arXiv:1506.07041v2 [math.PR]."},{"key":"ref011","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/jdt075"},{"key":"ref012","doi-asserted-by":"publisher","DOI":"10.1007\/s002850050148"},{"key":"ref013","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1019160258"},{"key":"ref014","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4471-3267-7"},{"key":"ref015","unstructured":"[15]A.Murray and T.Hunt, The Cell Cycle, Oxford Univ. 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