{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article13371","name":"Asymptotic Divergences and Strong Dichotomy","abstract":"The Schnorr-Stimm dichotomy theorem [Schnorr and Stimm, 1972] concerns finite-state gamblers that bet on infinite sequences of symbols taken from a finite alphabet \u03a3. The theorem asserts that, for any such sequence S, the following two things are true.\r\n(1) If S is not normal in the sense of Borel (meaning that every two strings of equal length appear with equal asymptotic frequency in S), then there is a finite-state gambler that wins money at an infinitely-often exponential rate betting on S.\r\n(2) If S is normal, then any finite-state gambler betting on S loses money at an exponential rate betting on S.\r\nIn this paper we use the Kullback-Leibler divergence to formulate the lower asymptotic divergence div(S||\u03b1) of a probability measure \u03b1 on \u03a3 from a sequence S over \u03a3 and the upper asymptotic divergence Div(S||\u03b1) of \u03b1 from S in such a way that a sequence S is \u03b1-normal (meaning that every string w has asymptotic frequency \u03b1(w) in S) if and only if Div(S||\u03b1)=0. We also use the Kullback-Leibler divergence to quantify the total risk Risk_G(w) that a finite-state gambler G takes when betting along a prefix w of S.\r\nOur main theorem is a strong dichotomy theorem that uses the above notions to quantify the exponential rates of winning and losing on the two sides of the Schnorr-Stimm dichotomy theorem (with the latter routinely extended from normality to \u03b1-normality). Modulo asymptotic caveats in the paper, our strong dichotomy theorem says that the following two things hold for prefixes w of S.\r\n(1') The infinitely-often exponential rate of winning in 1 is 2^{Div(S||\u03b1)|w|}.\r\n(2') The exponential rate of loss in 2 is 2^{-Risk_G(w)}.\r\nWe also use (1') to show that 1-Div(S||\u03b1)\/c, where c= log(1\/ min_{a\u2208\u03a3} \u03b1(a)), is an upper bound on the finite-state \u03b1-dimension of S and prove the dual fact that 1-div(S||\u03b1)\/c is an upper bound on the finite-state strong \u03b1-dimension of S.","keywords":["finite-state dimension","finite-state gambler","Kullback-Leibler divergence","normal sequences"],"author":[{"@type":"Person","name":"Huang, Xiang","givenName":"Xiang","familyName":"Huang","email":"mailto:huangx@iastate.edu","sameAs":"https:\/\/orcid.org\/0000-0002-4815-6130","affiliation":"Le Moyne College, Syracuse, NY 13214, USA","funding":"This research was supported in part by National Science Foundation Grants 1247051, 1545028, and 1900716."},{"@type":"Person","name":"Lutz, Jack H.","givenName":"Jack H.","familyName":"Lutz","email":"mailto:lutz@iastate.edu","affiliation":"Iowa State University, Ames, IA 50011, USA","funding":"This research was supported in part by National Science Foundation Grants 1247051, 1545028, and 1900716."},{"@type":"Person","name":"Mayordomo, Elvira","givenName":"Elvira","familyName":"Mayordomo","email":"mailto:elvira@unizar.es","affiliation":"Departamento de Inform\u00e1tica e Ingenier\u00eda de Sistemas, Instituto de Investigaci\u00f3n en Ingenier\u00eda de Arag\u00f3n, Universidad de Zaragoza, 50018 Zaragoza, Spain","funding":"Part of this work was supported by a grant from the Spanish Ministry of Science, Innovation and Universities (TIN2016-80347-R) and was partly done during a research stay at the Iowa State University supported by National Science Foundation Research Grant 1545028."},{"@type":"Person","name":"Stull, Donald M.","givenName":"Donald M.","familyName":"Stull","email":"mailto:dstull@iastate.edu","affiliation":"Iowa State University, Ames, IA 50011, USA","funding":"This research was supported in part by National Science Foundation Grants 1247051, 1545028, and 1900716."}],"position":51,"pageStart":"51:1","pageEnd":"51:15","dateCreated":"2020-03-04","datePublished":"2020-03-04","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Huang, Xiang","givenName":"Xiang","familyName":"Huang","email":"mailto:huangx@iastate.edu","sameAs":"https:\/\/orcid.org\/0000-0002-4815-6130","affiliation":"Le Moyne College, Syracuse, NY 13214, USA","funding":"This research was supported in part by National Science Foundation Grants 1247051, 1545028, and 1900716."},{"@type":"Person","name":"Lutz, Jack H.","givenName":"Jack H.","familyName":"Lutz","email":"mailto:lutz@iastate.edu","affiliation":"Iowa State University, Ames, IA 50011, USA","funding":"This research was supported in part by National Science Foundation Grants 1247051, 1545028, and 1900716."},{"@type":"Person","name":"Mayordomo, Elvira","givenName":"Elvira","familyName":"Mayordomo","email":"mailto:elvira@unizar.es","affiliation":"Departamento de Inform\u00e1tica e Ingenier\u00eda de Sistemas, Instituto de Investigaci\u00f3n en Ingenier\u00eda de Arag\u00f3n, Universidad de Zaragoza, 50018 Zaragoza, Spain","funding":"Part of this work was supported by a grant from the Spanish Ministry of Science, Innovation and Universities (TIN2016-80347-R) and was partly done during a research stay at the Iowa State University supported by National Science Foundation Research Grant 1545028."},{"@type":"Person","name":"Stull, Donald M.","givenName":"Donald M.","familyName":"Stull","email":"mailto:dstull@iastate.edu","affiliation":"Iowa State University, Ames, IA 50011, USA","funding":"This research was supported in part by National Science Foundation Grants 1247051, 1545028, and 1900716."}],"copyrightYear":"2020","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2020.51","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["https:\/\/doi.org\/10.1016\/S0890-5401(03)00187-1","http:\/\/arxiv.org\/abs\/1611.05911"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6357","volumeNumber":154,"name":"37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)","dateCreated":"2020-03-04","datePublished":"2020-03-04","editor":[{"@type":"Person","name":"Paul, Christophe","givenName":"Christophe","familyName":"Paul","email":"mailto:christophe.paul@lirmm.fr","sameAs":"https:\/\/orcid.org\/0000-0001-6519-975X","affiliation":"CNRS, Universit\u00e9 de Montpellier, France"},{"@type":"Person","name":"Bl\u00e4ser, Markus","givenName":"Markus","familyName":"Bl\u00e4ser","email":"mailto:mblaeser@cs.uni-saarland.de","affiliation":"Universit\u00e4t des Saarlandes, Saarbr\u00fccken, Germany"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article13371","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6357"}}}