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
References
1. Bari N.K., Trigonometric Series, Holt, Rinehart and Winston, New York, 1967.
2. Flajolet P., Sedgewick R., Analytic Combinatorics, Cambridge University Press, 2008.
3. Flajolet P., Gourdon X., Dumas P., “Mellin transforms and asymptotics: Harmonic sums”, Theoretical Computer Science, 144:1–2 (1995), 3–58.
4. Erd ̈os P., “On a Family of Symmetric Bernoulli Convolutions”, American Journal of Mathematics, 61:4 (1995), 974–976.
5. Erd ̈os P., “On the Smoothness Properties of a Family of Bernoulli Convolutions”, American Journal of Mathematics, 62:1 (1940), 180–186.
6. Garsia A.M., “Arithmetic Properties of Bernoulli Convolutions”, Transactions of the American Mathematical Society, 102:3 (1962), 409–432.
7. Jessen B., Wintner A., “Distribution Functions and the Riemann Zeta Function”, Transactions of the American Mathematical Society, 38:1 (1935), 48–88.
8. Peres Y., Schlag W., and Solomyak B., “Sixty years of Bernoulli convolutions”, Fractals and Stochastics II (C. Bandt, S. Graf and M. Zaehle, eds.), Birkhauser, 2000, 39–65.
9. Salem R., “Sets of Uniqueness and Sets of Multiplicity”, Transactions of the American Mathematical Society, 54:2 (1943), 218–228.
10. Salem R., “Sets of Uniqueness and Sets of Multiplicity. II”, Transactions of the American Mathematical Society, 56:1 (1944), 32–49.
11. Salem R., “Rectifications to the Papers Sets of Uniqueness and Sets of Multiplicity, I and II”, Transactions of the American Mathematical Society, 63:3 (1948), 595–598.
12. Solomyak B., “On the Random Series ±λn (an Erdos Problem)”, The Annals of Mathematics 2nd Ser., 142:3 (1995), 611–625.
13. Wintner A., “On Convergent Poisson Convolutions”, American Journal of Mathematics, 57:4 (1935), 827–838.
14. Szpankowski W., Average Case Analysis of Algorithms on Sequences, John Wiley & Sons, New York, 2001.
15. Gradstein I.S., Ryzhik I.M., Table of integrals, Series, and Products, Academic Press, 1994.
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
Views:
1087
Email this article 