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The law of large numbers and \(\sqrt{2}\). (English) Zbl 0823.60023
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.
60F15 Strong limit theorems
60F10 Large deviations
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