an:00751301
Zbl 0823.60023
Liggett, Thomas M.; Petersen, Peter
The law of large numbers and \(\sqrt{2}\)
EN
Am. Math. Mon. 102, No. 1, 31-35 (1995).
00024562
1995
j
60F15 60F10
law of large numbers; large deviations
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.
N.G.Gamkrelidze (Moskva)