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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Yaroslavl State University
2016-04-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/328 |
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| author | E. A. Timofeev |
| author_facet | E. A. Timofeev |
| author_sort | E. A. Timofeev |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-d056b59c9a1c4cec8d7b2bef02a0af52 |
| institution | DOAJ |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2016-04-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-d056b59c9a1c4cec8d7b2bef02a0af522025-08-20T03:22:04ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172016-04-0123218519410.18255/1818-1015-2016-2-185-194290Asymptotic formula for the moments of Bernoulli convolutionsE. A. Timofeev0P.G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, RussiaAbstract. 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.https://www.mais-journal.ru/jour/article/view/328momentsself-similarbernoulli convolutionsingularmellin transformasymptotic |
| spellingShingle | E. A. Timofeev Asymptotic formula for the moments of Bernoulli convolutions Моделирование и анализ информационных систем moments self-similar bernoulli convolution singular mellin transform asymptotic |
| title | Asymptotic formula for the moments of Bernoulli convolutions |
| title_full | Asymptotic formula for the moments of Bernoulli convolutions |
| title_fullStr | Asymptotic formula for the moments of Bernoulli convolutions |
| title_full_unstemmed | Asymptotic formula for the moments of Bernoulli convolutions |
| title_short | Asymptotic formula for the moments of Bernoulli convolutions |
| title_sort | asymptotic formula for the moments of bernoulli convolutions |
| topic | moments self-similar bernoulli convolution singular mellin transform asymptotic |
| url | https://www.mais-journal.ru/jour/article/view/328 |
| work_keys_str_mv | AT eatimofeev asymptoticformulaforthemomentsofbernoulliconvolutions |