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0225U004718,
(0125U001949)
,
Contribution to modern theory of random series
Граничні теореми для випадкових рядів, що задають майже періодичні випадкові процеси, та випадкових рядів, пов'язаних з випадковими еволюціями, що затухають. Точна асимптоматика хвостів розподілів випадкових рядів Діріхле
Iksanov Oleksandr М., Доктор фізико-математичних наук
17-12-2025
Taras Shevchenko National University of Kyiv
Invent novel probabilistic tools for solving several open problems for random series, both new and previously known. This programme should combine developing novel universal techniques and adaptation of some existing methods of Probability Theory and Number Theory, for instance, exponential change of measure, probabilistic local limit theorems and de Bruijn's argument of analyzing nontrivial functional equations arising in Number Theory.
The project is aimed at developing new methods that will allow solving a number of open problems for several types of random series. Our goal is to create a new probabilistic apparatus for solving a number of open problems for random series, both those first defined in this project and those studied earlier. This program should combine the development of new universal techniques and the adaptation of existing methods of probability theory and number theory, for example, exponential measure replacement, probabilistic local limit theorems and the de Bruyne technique for analyzing non-trivial functional equations inherent in number theory. The main tasks of the project: I) Prove limit theorems for random series that specify almost periodic random processes. II) Prove the repeated logarithm law for Dirichlet random series with explicitly given coefficients. III) Proving the repeated logarithm law for Dirichlet random series with implicitly given coefficients satisfying an asymptotic relation. Revealing the significance of this result for number theory. IV) Finding the exact first-order asymptotics (possibly higher orders) of the tails of the distributions of convergent Dirichlet random series. V) Finding the exact first-order asymptotics (possibly higher orders) of the tails of the distributions of convergent random series generated by linear recursions in the cases of superexponential and exponential tail decay. VI) Studying the properties of distributions and proving limit theorems for random series associated with decaying random evolutions and complex-valued random evolutions.
Samoilenko Ihor V.
Kostohryz Ruslan O.
Iksanov Oleh O.
Feshchenko Iryna O.
2025-12-17
Updated: 2025-12-17
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