Asymptotic formula for the moments of Bernoulli convolutions
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 calle...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2016-04-01
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| Series: | Modelirovanie i Analiz Informacionnyh Sistem |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/328 |
| Summary: | 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. |
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| ISSN: | 1818-1015 2313-5417 |