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Asymptotic Formula for the Moments of Lebesgue’s Singular Function

https://doi.org/10.18255/1818-1015-2015-5-723-730

Abstract

Recall Lebesgue’s singular function. Imagine flipping a biased coin with probability p of heads and probability q = 1 − p of tails. Let the binary expansion of ξ ∈ [0, 1]: ξ = ∑∞ k=1 ck2−k be determined by flipping the coin infinitely many times, that is, ck = 1 if the k-th toss is heads and ck = 0 if it is tails. We define Lebesgue’s singular function L(t) as the distribution function of the random variable ξ: L(t) = Prob{ξ < t}. It is well-known that L(t) is strictly increasing and its derivative is zero almost everywhere (p ̸= q). The moments of Lebesque’ singular function are defined as Mn = Eξn. The main result of this paper is the following: Mn = O(nlog2 p).

About the Author

E. A. Timofeev
Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia
Russian Federation

ScD, professor



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For citations:


Timofeev E.A. Asymptotic Formula for the Moments of Lebesgue’s Singular Function. Modeling and Analysis of Information Systems. 2015;22(5):723-730. (In Russ.) https://doi.org/10.18255/1818-1015-2015-5-723-730

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ISSN 1818-1015 (Print)
ISSN 2313-5417 (Online)