ASYMPTOTIC BEHAVIOR OF THE CHARACTERISTIC FUNCTION OF ONE DISTRIBUTION OF THE JESSEN-WINTNER TYPE

Abstract

The paper considers a random variable, which is the sum of a pointwise convergent random power series with independent discretely distributed terms that take on integer values. The corresponding random variable is a random variable represented by an s-fraction with a redundant set of digits and is included in the set of distributions of the Jessen-Wintner type. The Lebesgue distribution function of a random variable represented by an s-fraction with a redundant set of digits contains only a discrete or absolutely continuous or singular component. Emphasis in the paper is on the study of the asymptotic properties of the modulus of the characteristic function of a random variable represented by an s-fraction with a redundant set of digits. We consider the value $L$, which is the upper limit at infinity of the modulus of the characteristic function of the corresponding random variable. The value $L$ being equal to one and zero for a discrete and absolutely continuous distribution, respectively, can acquire an arbitrary predetermined value from the segment $[0;1]$ for a singular distribution. $L$ is a measure of closeness to a discrete, absolutely continuous or singular distribution. Calculating exact values $L$ or their estimation for singular distributions is a non-trivial, complex task. In the work, the necessary and sufficient conditions for the equality of the value of the upper bound at infinity to the modulus of the characteristic function of the corresponding random variable, under certain asymptotic restrictions, were found. The limit ratios $L$ for the calculation are indicated, in particular it is shown that the value $L$ is the limit value of a certain subsequence of modules of the Fourier-Stiltjes coefficients.