Abstract
The asymptotic behavior of the modulus of a characteristic function of a random variable, the distribution function of which is the classical singular Cantor function, is investigated. The emphasis is on calculating the upper bound of the modulus of the characteristic Cantor distribution function. The probabilistic measure corresponding to Cantor's distribution belongs to the class of Bernoulli's symmetric convolutions, the interest in which is considerable today. Bernoulli's symmetrical convolutions were actively studied by both domestic mathematicians: M. Pratsovyty, G. Turbin, G. Torbin, J. Honcharenko, O. Baranovsky and others, and foreign ones: Erdos P, Peres Y, Schlag W, Solomyak B, Albeverio, S and other. The value of the upper bound of the modulus of the characteristic function plays an important role in the problem of determining the Lebesgue structure of distributions of sums of probably convergent random series with independent discrete terms (random values of the Jessen-Winter type). The exact value of the upper bound of the module of the characteristic Cantor distribution function is found in the article.
Highlights
В роботi дослiджується асимптотична поведiнка модуля характеристичної функцiї випадкової величини, функцiєю розподiлу якої являється класична сингулярна функцiя Кантора
The emphasis is on calculating the upper bound of the modulus of the characteristic Cantor distribution function
The value of the upper bound of the modulus of the characteristic function plays an important role in the problem of determining the Lebesgue structure of distributions of sums of probably convergent random series with independent discrete terms
Summary
В роботi дослiджується асимптотична поведiнка модуля характеристичної функцiї випадкової величини, функцiєю розподiлу якої являється класична сингулярна функцiя Кантора. Асимптотична поведiнка модуля характеристичної функцiї розподiлу Кантора 1 Центральноукраїнський державний педагогiчний унiверситет iменi Володимира Винниченка, 25006, м. Приклад характеристичної функцiї сингулярного розподiлу ψ, для якого Lψ = 1 наведено Вiнтнером в роботi [10] та Єссеєном в [7]. Нехай πtn — зростаюча необмежена послiдовнiсть дiйсних чисел така, що lim n→+∞
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