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Model. Anal. Inform. Sist., 2016, Volume 23, Number 2, Pages 185–194 (Mi mais490)  

Asymptotic formula for the moments of Bernoulli convolutions

E. A. Timofeev

P.G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia

Abstract: For each $\lambda$, $0<\lambda<1$, we define a random variable
$$ Y_\lambda = (1-\lambda)\sum_{n=0}^\infty \xi_n\lambda^n, $$
where $\xi_n$ are independent random variables with
$$ \mathrm{P}\{\xi_n =0\} =\mathrm{P}\{\xi_n =1\} =\frac12. $$
The distribution of $Y_\lambda$ is called a symmetric Bernoulli convolution. The main result of this paper is
$$ M_n = \mathrm{E} Y_\lambda^n = n^{\log_{\lambda}2} 2^{\log_\lambda(1-\lambda)+0.5\log_\lambda2-0.5} e^{\tau(-\log_{\lambda}n)}(1 + \mathcal{O}(n^{-0.99})), $$
where
$$ \tau(x)=\sum_{k\ne0}\frac1k\alpha(-\frac{k}{\ln\lambda})e^{2\pi ikx} $$
is a 1-periodic function,
$$ \alpha(t) = -\frac{1}{2i\mathrm{sh} (\pi^2t)} (1-\lambda)^{2\pi i t}(1 - 2^{2\pi i t})\pi^{-2\pi i t }2^{-2\pi i t }\zeta(2\pi i t), $$
and $\zeta(z)$ is the Riemann zeta function.
The article is published in the author's wording.

Keywords: moments, self-similar, Bernoulli convolution, singular, Mellin transform, asymptotic.

DOI: https://doi.org/10.18255/1818-1015-2016-2-185-194

Full text: PDF file (625 kB)
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Bibliographic databases:

UDC: 519.987
Received: 08.02.2016
Language:

Citation: E. A. Timofeev, “Asymptotic formula for the moments of Bernoulli convolutions”, Model. Anal. Inform. Sist., 23:2 (2016), 185–194

Citation in format AMSBIB
\Bibitem{Tim16}
\by E.~A.~Timofeev
\paper Asymptotic formula for the moments of Bernoulli convolutions
\jour Model. Anal. Inform. Sist.
\yr 2016
\vol 23
\issue 2
\pages 185--194
\mathnet{http://mi.mathnet.ru/mais490}
\crossref{https://doi.org/10.18255/1818-1015-2016-2-185-194}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3504588}
\elib{http://elibrary.ru/item.asp?id=25810351}


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