{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:35:05Z","timestamp":1759970105671,"version":"build-2065373602"},"reference-count":6,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,3]],"date-time":"2025-01-03T00:00:00Z","timestamp":1735862400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"TU Wien Bibliothek"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Consider a coin-tossing sequence, i.e., a sequence of independent variables, taking values 0 and 1 with probability 1\/2. The famous Erd\u0151s-R\u00e9nyi law of large numbers implies that the longest run of ones in the first n observations has a length Rn that behaves like log(n), as n tends to infinity (throughout this article, log denotes logarithm with base 2). Erd\u0151s and R\u00e9v\u00e9sz refined this result by providing a description of the L\u00e9vy upper and lower classes of the process Rn. In another direction, Arratia and Waterman extended the Erd\u0151s-R\u00e9nyi result to the longest matching subsequence (with shifts) of two coin-tossing sequences, finding that it behaves asymptotically like 2log(n). The present paper provides some Erd\u0151s-R\u00e9v\u00e9sz type results in this situation, obtaining a complete description of the upper classes and a partial result on the lower ones.<\/jats:p>","DOI":"10.3390\/e27010034","type":"journal-article","created":{"date-parts":[[2025,1,3]],"date-time":"2025-01-03T07:10:17Z","timestamp":1735888217000},"page":"34","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Erd\u0151s-R\u00e9v\u00e9sz Type Law for the Length of the Longest Match of Two Coin-Tossing Sequences"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5674-379X","authenticated-orcid":false,"given":"Karl","family":"Grill","sequence":"first","affiliation":[{"name":"Institute of Statistics and Mathematical Methods in Economy, TU Wien, Wiedner Hauptstra\u00dfe 8-10, 1040 Vienna, Austria"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/BF02795493","article-title":"On a new law of large numbers","volume":"23","year":"1970","journal-title":"J. Anal. Math."},{"key":"ref_2","unstructured":"Cisz\u00e1r, I., and Elias, P. (1977). On the length of the longest head-run. Topics in Information Theory, North-Holland. Colloquia Mathematica Societatis Janos Bolyai Volume, 16."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/0001-8708(85)90003-9","article-title":"An Erd\u0151s-R\u00e9nyi law with shifts","volume":"55","author":"Arratia","year":"1985","journal-title":"Adv. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/BF01955038","article-title":"Large deviation results for waiting times in repeated experiments","volume":"45","year":"1985","journal-title":"Acta Math. Hung."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1016\/0167-7152(84)90028-2","article-title":"Asymptotic independence of pure head stopping times","volume":"2","year":"1984","journal-title":"Stat. Probab. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/BF01312212","article-title":"On the waiting time till each of some given patterns occurs as a run","volume":"87","year":"1991","journal-title":"Probab. Theory Relat. Fields"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/1\/34\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:22:26Z","timestamp":1759918946000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/1\/34"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,3]]},"references-count":6,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["e27010034"],"URL":"https:\/\/doi.org\/10.3390\/e27010034","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,1,3]]}}}