Asymptotic formula for the moments of Lebesgue’s singular function

E. A. Timofeev

Modelirovanie i analiz informacionnyh sistem, Tome 22 (2015) no. 5, pp. 723-730

Résumé

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 $\xi\in[0,1]$: $ \xi = \sum_{k=1}^{\infty}c_k2^{-k}$ be determined by flipping the coin infinitely many times, that is, $c_k =1$ if the $k$-th toss is heads and $c_k =0$ if it is tails. We define Lebesgue's singular function $L(t)$ as the distribution function of the random variable $\xi$: $$ L(t) = Prob\{\xi t\}. $$ It is well-known that $L(t)$ is strictly increasing and its derivative is zero almost everywhere ($p\ne q$). The moments of Lebesque' singular function are defined as $$ M_n = \mathsf{E}\xi^n. $$ The main result of this paper is the following: $$ M_n = O(n^{\log_2 p}). $$

Détail

Mots-clés : moments, Lebesgue’s function
Keywords: self-similar, singular, Mellin transform, asymptotic.