Related papers: Summation arithmetic functions with bounded terms,…
The aim of the paper is to study the limit distributions and the asymptotic behavior of summation arithmetic functions. A probabilistic approach based on the use of the axioms of probability theory is used for these purposes. Sufficient…
Summation arithmetic functions with asymptotically independent terms are studied in the paper, the limit of which is the law of normal distribution. Assertions about the asymptotic behavior of the indicated functions are proved.
The summation arithmetic functions with asymptotically independent summands are studied in the paper. We prove statements about the condition under which the summation arithmetic functions have asymptotically independent summands. It is…
The paper compares probabilistic and exact methods for estimating the asymptotic behavior of summation arithmetic functions, and estimates of the results are obtained by precise methods. Conditions for stationarity in the broad sense are…
For nonstationary, strongly mixing sequences of random variables taking their values in a finite-dimensional Euclidean space, with the partial sums being normalized via matrix multiplication, with certain standard conditions being met, the…
This paper proves several assertions on sufficient conditions for the convergence of additive arithmetic functions to the normal distribution. A generalization of the Erdos-Kac theorem was proved and determines the rate of convergence of…
The paper considers estimates for the asymptotics of summation functions of bounded multiplicative arithmetic functions. Several assertions on this subject are proved and examples are considered.
Summation arithmetic functions of Mertens and Liouville are investigated in the paper. It is proved that the limiting distribution of these functions is the normal. It is also shown that the estimating of standard deviation of these…
In this article we introduce associative Look-Up Tables. With their help, pseudo sums are correctly determined. The set of limit distributions in a pseudo-summation scheme of i.i.d. random variables is described. Also, two special cases…
In applied probability, the normal approximation is often used for the distribution of data with assumed additive structure. This tradition is based on the central limit theorem for sums of (independent) random variables. However, it is…
We consider random multiplicative functions taking the values $\pm 1$. Using Stein's method for normal approximation, we prove a central limit theorem for the sum of such multiplicative functions in appropriate short intervals.
For a stationary sequence that is regularly varying and associated we give conditions which guarantee that partial sums of this sequence, under normalization related to the exponent of regular variation, converge in distribution to a…
An approach will be proposed to determine the existence of a limit distribution of additive arithmetic functions in this work. It is based on assertions that will be proven in this work and on the properties of Dirichlet convolution and…
We present new mixture representations for the generalized Linnik distribution in terms of normal, Laplace, exponential and stable laws and establish the relationship between the mixing distributions in these representations. Based on these…
We consider moderately trimmed sums of non-negative i.i.d. random variables. We show that for every distribution function there exists a proper moderate trimming such that for the trimmed sum a non-trivial strong law of large numbers holds.…
The aim of this paper is to provide conditions which ensure that the affinely transformed partial sums of a strictly stationary process converge in distribution to an infinite variance stable distribution. Conditions for this convergence to…
Motivated by questions arising in the study of the spectral theory of models of aperiodic order, we investigate sums of functions of semibounded closed subsets of the real line. We show that under suitable thickness assumptions on the sets…
Several properties of stationary subdivision schemes are nowadays well understood. In particular, it is known that the polynomial generation and reproduction capability of a stationary subdivision scheme is strongly connected with sum…
We consider the summatory function of the totient function after applications of a suitable smoothing operator and study the limiting behavior of the associated error term. Under several conditional assumptions, we show that the smoothed…
We study the asymptotics of the moments of arithmetic functions that have a limit distribution, not necessarily normal, defined on a subset of the natural series that satisfies certain requirements. Several assertions are proved on…