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Teoriya Veroyatnostei i ee Primeneniya, 1978, Volume 23, Issue 3, Pages 540–547
(Mi tvp3074)
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This article is cited in 4 scientific papers (total in 5 papers)
A local limit theorem for the distribution of fractional parts of an exponential function
D. A. Moskvin, A. G. Postnikov Moscow
Abstract:
Let $\Delta=[\alpha,\beta]\subset[0,1]$, $|\Delta|=\beta-\alpha$. Let $\chi(t)$ be the indicator function of $\Delta$. The number of fractional parts of $\{\xi2^x\}$, $x=0,1,\dots,n-1$, belonging to the segment $\Delta$ is
$$
N_n(\xi,\Delta)=\sum_{x=0}^n \chi(\{\xi2^x\}).
$$
Let $\tau$ be a non-negative integer number. The following result is proved:
Theorem. For $n\to\infty$, uniformly in $\tau$,
$$
\operatorname{mes}\{\xi:0\le\xi\le 1,\,N_n(\xi,\Delta)=\tau\}=
\frac{1}{\sigma\sqrt{2\pi n}}\exp
\biggl(-\frac{(\tau-n|\Delta|)^2}{2n\sigma^2}\biggr)+O\biggl(\frac{\sqrt{\ln n}}{n}\biggr).
$$
Received: 26.07.1976
Citation:
D. A. Moskvin, A. G. Postnikov, “A local limit theorem for the distribution of fractional parts of an exponential function”, Teor. Veroyatnost. i Primenen., 23:3 (1978), 540–547; Theory Probab. Appl., 23:3 (1979), 521–528
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https://www.mathnet.ru/eng/tvp3074 https://www.mathnet.ru/eng/tvp/v23/i3/p540
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