Asymptotic formula for the moments of Bernoulli convolutions
https://doi.org/10.18255/1818-1015-2016-2-185-194
Abstract
Abstract.
Asymptotic Formula for the Moments of Bernoulli Convolutions
Timofeev E. A.
Received February 8, 2016
For each λ, 0 < λ < 1, we define a random variable
∞
Yλ =(1−λ)ξnλn, n=0
where ξn are independent random variables with
P{ξn =0}=P{ξn =1}= 1.
2
The distribution of Yλ is called a symmetric Bernoulli convolution. The main result of this paper is Mn =EYλn =nlogλ22logλ(1−λ)+0.5logλ2−0.5eτ(−logλn)1+O(n−0.99),
where
is a 1-periodic function,
1k2πikx τ(x)= kα −lnλ e
k̸=0
1 (1 − λ)2πit(1 − 22πit)π−2πit2−2πitζ(2πit), 2i sh(π2t)
α(t) = −
and ζ(z) is the Riemann zeta function.
The article is published in the author’s wording.
About the Author
E. A. Timofeev
P.G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia
Russian Federation
Russian Federation
ScD, professor
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Review
For citations:
Timofeev E.A. Asymptotic formula for the moments of Bernoulli convolutions. Modeling and Analysis of Information Systems. 2016;23(2):185-194. https://doi.org/10.18255/1818-1015-2016-2-185-194
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