Abstract
We consider an analog of the problem of the existence of the summable distributional density of a random variable in the form of power series on the dyadic half-line which was originally proposed and partially solved by Erdös on the standard real line. Given a random variable \( \xi \) as a series of the powers of \( \lambda\in(0,1) \), we address the question of \( \lambda \) such that the density of \( \xi \) belongs to the space of the function whose modulus is summable on the dyadic half-line. We answer the question for some values of \( \lambda \), and consider the so-called dual problem when \( \lambda=\frac{1}{2} \) is fixed, but the coefficients of the formula for \( \xi \) have more degrees of freedom. Also we obtain some criteria for the existence of density in terms of the solution of the refinement equation tied directly to \( \xi \) as well as in terms of the coefficients defining \( \xi \).
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Acknowledgments
The author expresses his deepest gratitude to Professor Vladimir Yu. Protasov and the anonymous reviewer for the valuable comments and discussion of the work.
Funding
This work is supported by Grant no. 075–02–2023–924.
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Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 6, pp. 1186–1198. https://doi.org/10.33048/smzh.2023.64.607
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Karapetyants, M.A. On the Distribution of a Random Power Series on the Dyadic Half-Line. Sib Math J 64, 1319–1329 (2023). https://doi.org/10.1134/S0037446623060071
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DOI: https://doi.org/10.1134/S0037446623060071
Keywords
- dyadic half-line
- random variable
- distributional density
- power series
- Walsh functions
- Walsh-Fourier transform
- refinement equation