Asymptotic Formula for the Moments of Lebesgue’s Singular Function
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...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2015-10-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/287 |
| _version_ | 1826559002008354816 |
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| author | E. A. Timofeev |
| author_facet | E. A. Timofeev |
| author_sort | E. A. Timofeev |
| collection | DOAJ |
| description | 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). |
| first_indexed | 2024-04-10T02:24:27Z |
| format | Article |
| id | doaj.art-f57377fd6d2f429fab9bf549f7ad59ad |
| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2025-03-14T08:53:27Z |
| publishDate | 2015-10-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-f57377fd6d2f429fab9bf549f7ad59ad2025-03-02T12:46:57ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172015-10-0122572373010.18255/1818-1015-2015-5-723-730268Asymptotic Formula for the Moments of Lebesgue’s Singular FunctionE. A. Timofeev0Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, RussiaRecall 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).https://www.mais-journal.ru/jour/article/view/287momentsself-similarlebesgue’s functionsingularmellin transformasymptotic |
| spellingShingle | E. A. Timofeev Asymptotic Formula for the Moments of Lebesgue’s Singular Function Моделирование и анализ информационных систем moments self-similar lebesgue’s function singular mellin transform asymptotic |
| title | Asymptotic Formula for the Moments of Lebesgue’s Singular Function |
| title_full | Asymptotic Formula for the Moments of Lebesgue’s Singular Function |
| title_fullStr | Asymptotic Formula for the Moments of Lebesgue’s Singular Function |
| title_full_unstemmed | Asymptotic Formula for the Moments of Lebesgue’s Singular Function |
| title_short | Asymptotic Formula for the Moments of Lebesgue’s Singular Function |
| title_sort | asymptotic formula for the moments of lebesgue s singular function |
| topic | moments self-similar lebesgue’s function singular mellin transform asymptotic |
| url | https://www.mais-journal.ru/jour/article/view/287 |
| work_keys_str_mv | AT eatimofeev asymptoticformulaforthemomentsoflebesguessingularfunction |