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. TimofeevRussian Federation
ScD, professor
References
1. Flajolet P., Sedgewick R., Analytic Combinatorics, Cambridge University Press, 2008.
2. Lomnicki Z., Ulam S. E., “Sur la theorie de la mesure dans les espaces combinatoires et son application au calcul des probabilites. I. Variables independantes”, Fundamenta Mathematicae, 23:1 (1934), 237–278.
3. Salem R., “On some singular monotonic functions which are strictly increasing,”, Trans. Amer. Math. Soc., 53:3 (1943), 427–439.
4. De Rham G., “On Some Curves Defined by Functional Equations”, Classics on Fractals, ed. Gerald A. Edgar (Ed.), Addison-Wesley, 1993, 285–298.
5. Szpankowski W.,, Average Case Analysis of Algorithms on Sequences, John Wiley & Sons, New York, 2001.
6. Gradstein I. S., Ryzhik I. M., Table of integrals, Series, and Products, Academic Press, 1994.
7. Timofeev E. A., “Bias of a nonparametric entropy estimator for Markov measures”, Journal of Mathematical Sciences, 176:2 (2011), 255–269.
Review
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