Related papers: Additive arithmetic functions with limit normal di…
We establish asymptotic formulae for various correlations involving general divisor functions $d_k(n)$ and partial divisor functions $d_l(n,A)=\sum_{q|n:q\leq n^A}d_{l-1}(q)$, where $A\in[0,1]$ is a parameter and $k,l\in\mathbb{N}$ are…
We establish some limit theorems for quasi-arithmetic means of random variables. This class of means contains the arithmetic, geometric and harmonic means. Our feature is that the generators of quasi-arithmetic means are allowed to be…
In this short note we study uniform approximations to the normal distributions by Jacobi theta functions. We shall show that scaled theta functions approach to a normal distribution exponentially fast.
In this technical report, certain interesting classification of arithmetical functions is proposed. The notion of additively decomposable and multiplicatively decomposable arithmetical functions is proposed. The concepts of arithmetical…
We study the concentration of the distribution of an additive function, when the sequence of prime values of $f$ decays fast and has good spacing properties. In particular, we prove a conjecture by Erdos and Katai on the concentration of…
We prove an extension of the Regularity Lemma with vertex and edge weights which can be applied for a large class of graphs. The applications involve random graphs and a weighted version of the Erd\H{o}s-Stone theorem. We also provide means…
The almost sure convergence of ergodic averages in Birkhoff's pointwise ergodic theorem is known to fail in the finitely additive setting. We introduce a natural reformulation of almost sure convergence suitable for finitely additive…
Consider d uniformly random permutation matrices on n labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices. We…
The assumption of normality in data has been considered in the field of statistical analysis for a long time. However, in many practical situations, this assumption is clearly unrealistic. It has recently been suggested that the use of…
We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.
A characterization of multiplicative (and additive) arithmetical functions is given. Using this characterization, we show that the group of multiplicative arithmetical functions is isomorphic to the group of additive arithmetical functions.
By using limit theorems of uniform mixing Markov processes and martingale difference sequences, the strong law of large numbers, central limit theorem, and the law of iterated logarithm are established for additive functionals of…
The paper is a sketch of systematic presentation of distributional limit theorems and their refinements for compound sums. When analyzing, e.g., ergodic semi-Markov systems with discrete or continuous time, this allows us to separate those…
We estimate the average of any arithmetic function $k$ over the values of any smooth polynomial in many variables provided only that $k$ has a distribution in arithmetic progressions of fixed modulus. We give several applications of this…
We prove a Central Limit Theorem for probability measures defined via the variation of the sum-of-digits function, in base $b\ge 2$. For $r\ge 0$ and $d \in \mathbb{Z}$, we consider $\mu^{(r)}(d)$ as the density of integers $n\in…
The Central Limit Theorem states that, in the limit of a large number of terms, an appropriately scaled sum of independent random variables yields another random variable whose probability distribution tends to a stable distribution. The…
The purpose of this paper is to study the limiting distribution of special {\it additive functionals} on random planar maps, namely the number of occurrences of a given {\it pattern}. The main result is a central limit theorem for these…
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…
Correction to Annals of Probability 28 (2000) 277--302 [doi:10.1214/aop/1019160120].
Let $g \geq 2$. A real number is said to be g-normal if its base g expansion contains every finite sequence of digits with the expected limiting frequency. Let \phi denote Euler's totient function, let \sigma be the sum-of-divisors…