Asymptotic Formula for the Moments of Lebesgue’s Singular Function

• Published in: Modelirovanie i analiz informacionnyh sistem Vol. 22; no. 5; pp. 723 - 730
• Main Author: Timofeev, E. A.
• Format: Journal Article
• Language: English
• Published: Yaroslavl State University 04.12.2015
• Subjects: asymptotic; lebesgue’s function; mellin transform; moments; self-similar; singular
• ISSN: 1818-1015; 2313-5417
• DOI: 10.18255/1818-1015-2015-5-723-730

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).