{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T04:22:29Z","timestamp":1759638149744},"reference-count":29,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2015,7,7]],"date-time":"2015-07-07T00:00:00Z","timestamp":1436227200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2016,5]]},"abstract":"<jats:p>In this note we establish a uniform bound for the distribution of a sum <jats:italic>S<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>=<jats:italic>X<\/jats:italic><jats:sub>1<\/jats:sub>+\u00b7\u00b7\u00b7+<jats:italic>X<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> of independent non-homogeneous Bernoulli trials. Specifically, we prove that \u03c3<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub><jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548315000127_char1\" \/><\/jats:private-char>(<jats:italic>S<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> = <jats:italic>j<\/jats:italic>) \u2264 \u03b7, where \u03c3<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> denotes the standard deviation of <jats:italic>S<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>, and \u03b7 is a universal constant. We compute the best possible constant \u03b7 ~ 0.4688 and we show that the bound also holds for limits of sums and differences of Bernoullis, including the Poisson laws which constitute the worst case and attain the bound. We also investigate the optimal bounds for <jats:italic>n<\/jats:italic> and <jats:italic>j<\/jats:italic> fixed. An application to estimate the rate of convergence of Mann's fixed-point iterations is presented.<\/jats:p>","DOI":"10.1017\/s0963548315000127","type":"journal-article","created":{"date-parts":[[2015,7,7]],"date-time":"2015-07-07T04:40:15Z","timestamp":1436244015000},"page":"352-361","source":"Crossref","is-referenced-by-count":9,"title":["A Sharp Uniform Bound for the Distribution of Sums of Bernoulli Trials"],"prefix":"10.1017","volume":"25","author":[{"given":"JEAN-BERNARD","family":"BAILLON","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ROBERTO","family":"COMINETTI","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"JOS\u00c9","family":"VAISMAN","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2015,7,7]]},"reference":[{"key":"S0963548315000127_ref23","doi-asserted-by":"publisher","DOI":"10.1137\/1124073"},{"key":"S0963548315000127_ref25","doi-asserted-by":"publisher","DOI":"10.1137\/1106009"},{"key":"S0963548315000127_ref20","first-page":"123","article-title":"Two remarks on the method of successive approximations.","volume":"10","author":"Krasnosel'ski","year":"1955","journal-title":"Uspekhi Mat. 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