Liggett, Thomas M.; Petersen, Peter The law of large numbers and \(\sqrt{2}\). (English) Zbl 0823.60023 Am. Math. Mon. 102, No. 1, 31-35 (1995). Let \(b_ n\) be a positive sequence which satisfies \[ \lim_{n \to \infty} \root n \of {b_ n} \lambda > 0,\tag{*} \] where \(b^{(k)}_ n = b^{(k -1)}_ n + b^{(k - 1)}_ n\) and \(b^{(0)}_ n = b_ n\). It is not too hard to observe that if \(b^{(0)}_ n = (1,1,2,2,4,4,\dots)\), then \(b^{(k)}_ 1 /b^{(k)}_ 0\) approximates \(\sqrt{2}\). The authors are interested in seeing how generally this procedure works. They show that under the above condition \((*)\) it is true \(\lim_{k \to \infty} b^{(k)}_ 1 / b^{(k)}_ 0 = \lambda\). The method of proof is based on an elementary version of the law of large numbers and a strengthened form of large deviations. Reviewer: N.G.Gamkrelidze (Moskva) MSC: 60F15 Strong limit theorems 60F10 Large deviations Keywords:law of large numbers; large deviations PDFBibTeX XMLCite \textit{T. M. Liggett} and \textit{P. Petersen}, Am. Math. Mon. 102, No. 1, 31--35 (1995; Zbl 0823.60023) Full Text: DOI