Asymptotic formula for the moments of Bernoulli convolutions

Modelirovanie i Analiz Informacionnyh Sistem

E. A. Timofeev¹ 

[1] P.G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia;

关键词: moments;    self-similar;    bernoulli convolution;    singular;    mellin transform;    asymptotic;   

DOI  :  10.18255/1818-1015-2016-2-185-194

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown