mathnasТруды Института математики НАН БеларусиProceedings of the Institute of Mathematics of the NAS of Belarus1812-5093Институт математики НАН БеларусиRXOTNQmathnas-98Research ArticleВЕЩЕСТВЕННЫЙ, КОМПЛЕКСНЫЙ И ФУНКЦИОНАЛЬНЫЙ АНАЛИЗREAL, COMPLEX AND FUNCTIONAL ANALYSISСвойство RUC для хаоса случайных величин в равномерной нормеRUC property for chaos of random variables in the uniform normСлиняковП. А.SlinyakovP. A.

Минск

Minsk

slinakovp@gmail.comЛыковК. В.LykovK. V.

Минск

Minsk

alkv@list.ruИнститут математики НАН Беларуси; Белорусский государственный университетInstitute of Mathematics of the National Academy of Sciences of Belarus; Belarusian State University2025050120263325472Copyright © Слиняков П.А., Лыков К.В., 20262026Слиняков П.А., Лыков К.В.Slinyakov P.A., Lykov K.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://mathnas.ejournal.by/jour/article/view/98

Пусть $X=\{X_k\}_{k=1}^\infty$ – последовательность независимых симметричных и ограниченных случайных величин. В работе рассматриваются системы вида $\{X_iX_j\}_{i<j}$, $\{X_i X_j X_k\}_{i<j<k},\ldots$, конечные объединения таких систем и близкие к ним системы в пространстве $L_\infty$ ограниченных случайных величин. Ряды по таким системам не обладают свойством безусловности: сходимость рядов зависит от порядка, в котором нумеруются элементы системы. В то же время, как показано в работе, такие системы обладают очень бизким свойством случайной безусловной сходимости.}

Let $X=\{X_k\}_{k=1}^\infty$ be a sequence of independent symmetric bounded random variables. This paper investigates systems of the form $\{X_iX_j\}_{i<j}$, $\{X_i X_j X_k\}_{i<j<k},\ldots$, finite unions of such systems, and systems close to them, in the space $L_\infty$ of bounded random variables. Series over such systems do not hold the property of unconditionality: the convergence of the series depends on the ordering of the terms. At the same time, as we demonstrate in the paper, such systems posess a very close property of random unconditional convergence (or RUC-property).

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The authors declare that there are no conflicts of interest present.