maisМоделирование и анализ информационных системModeling and Analysis of Information Systems1818-10152313-5417Yaroslavl State University10.18255/1818-1015-2016-2-185-194mais-328Research ArticleОригинальные статьиArticlesАсимптотика моментов симметричной свертки БернуллиAsymptotic formula for the moments of Bernoulli convolutionsTимофеевЕ. А.TimofeevE. A.

доктор. физ.-мат. наук, профессор кафедры теоретической информатики 

ScD, professor 

timofeevEA@gmail.comЯрославский государственный университет им. П.Г. Демидова, ул. Советская, 14, г. Ярославль, 150000 РоссияРоссияP.G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, RussiaRussian Federation201620042016232185194Copyright © Tимофеев Е.А., 20162016Tимофеев Е.А.Timofeev E.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.mais-journal.ru/jour/article/view/328

Для каждого λ, 0 < λ < 1 определим случайную величину (симметричную свертку Бернулли) где ξn – независимые случайные величины с P{ξn =0}=P{ξn =1}= 1. ∞ Yλ =(1−λ)ξnλn, n=0 2 Mn =EYλn =nlogλ22logλ(1−λ)+0.5logλ2−0.5eτ(−logλn)1+O(n−0.99), 1k2πikx τ(x)= kα −lnλ e Основной результат настоящей работы где функция k̸=0 является периодической с периодом равным 1, α(t) = − 1 (1 − λ)2πit(1 − 22πit)π−2πit2−2πitζ(2πit), 2i sh(π2t) а ζ(z) – дзета-функция Римана.

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.

моментысамоподобиесвертка Бернуллисингулярная функцияпреобразование Меллинаасимптотикаmomentsself-similarBernoulli convolutionsingularMellin transformasymptoticReferencesBari N.K., Trigonometric Series, Holt, Rinehart and Winston, New York, 1967.Bari N.K., Trigonometric Series, Holt, Rinehart and Winston, New York, 1967.Flajolet P., Sedgewick R., Analytic Combinatorics, Cambridge University Press, 2008.Flajolet P., Sedgewick R., Analytic Combinatorics, Cambridge University Press, 2008.Flajolet P., Gourdon X., Dumas P., “Mellin transforms and asymptotics: Harmonic sums”, Theoretical Computer Science, 144:1–2 (1995), 3–58.Flajolet P., Gourdon X., Dumas P., “Mellin transforms and asymptotics: Harmonic sums”, Theoretical Computer Science, 144:1–2 (1995), 3–58.Erd ̈os P., “On a Family of Symmetric Bernoulli Convolutions”, American Journal of Mathematics, 61:4 (1995), 974–976.Erd ̈os P., “On a Family of Symmetric Bernoulli Convolutions”, American Journal of Mathematics, 61:4 (1995), 974–976.Erd ̈os P., “On the Smoothness Properties of a Family of Bernoulli Convolutions”, American Journal of Mathematics, 62:1 (1940), 180–186.Erd ̈os P., “On the Smoothness Properties of a Family of Bernoulli Convolutions”, American Journal of Mathematics, 62:1 (1940), 180–186.Garsia A.M., “Arithmetic Properties of Bernoulli Convolutions”, Transactions of the American Mathematical Society, 102:3 (1962), 409–432.Garsia A.M., “Arithmetic Properties of Bernoulli Convolutions”, Transactions of the American Mathematical Society, 102:3 (1962), 409–432.Jessen B., Wintner A., “Distribution Functions and the Riemann Zeta Function”, Transactions of the American Mathematical Society, 38:1 (1935), 48–88.Jessen B., Wintner A., “Distribution Functions and the Riemann Zeta Function”, Transactions of the American Mathematical Society, 38:1 (1935), 48–88.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.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.Salem R., “Sets of Uniqueness and Sets of Multiplicity”, Transactions of the American Mathematical Society, 54:2 (1943), 218–228.Salem R., “Sets of Uniqueness and Sets of Multiplicity”, Transactions of the American Mathematical Society, 54:2 (1943), 218–228.Salem R., “Sets of Uniqueness and Sets of Multiplicity. II”, Transactions of the American Mathematical Society, 56:1 (1944), 32–49.Salem R., “Sets of Uniqueness and Sets of Multiplicity. II”, Transactions of the American Mathematical Society, 56:1 (1944), 32–49.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.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.Solomyak B., “On the Random Series ±λn (an Erdos Problem)”, The Annals of Mathematics 2nd Ser., 142:3 (1995), 611–625.Solomyak B., “On the Random Series ±λn (an Erdos Problem)”, The Annals of Mathematics 2nd Ser., 142:3 (1995), 611–625.Wintner A., “On Convergent Poisson Convolutions”, American Journal of Mathematics, 57:4 (1935), 827–838.Wintner A., “On Convergent Poisson Convolutions”, American Journal of Mathematics, 57:4 (1935), 827–838.Szpankowski W., Average Case Analysis of Algorithms on Sequences, John Wiley & Sons, New York, 2001.Szpankowski W., Average Case Analysis of Algorithms on Sequences, John Wiley & Sons, New York, 2001.Gradstein I.S., Ryzhik I.M., Table of integrals, Series, and Products, Academic Press, 1994.Gradstein I.S., Ryzhik I.M., Table of integrals, Series, and Products, Academic Press, 1994.

The authors declare that there are no conflicts of interest present.