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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.
##### MSC:
 60F15 Strong limit theorems 60F10 Large deviations
##### Keywords:
law of large numbers; large deviations
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