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Vytas Zacharovas

Publications and source records attributed to Vytas Zacharovas.

16 recordsLinked to original sources
arXiv2026

Elementary asymptotics for the Stirling numbers of the second kind: The central range

We derive the local and central limit theorems for the Stirling numbers of the second kind by elementary means, obtaining as corollaries effective asymptotic estimates for the Bell numbers and for the moments of the distribution. We also develop asymptotic expansions along several directions, all based on a novel finite-differencing approach; this provides the first self-contained elementary justification of such expansions.

math.CO↗
arXiv2025

An Elementary Approach to Depoissonization

We study depoissonization for sequences with entire exponential generating functions. We establish explicit finite-order remainder bounds for Poisson--Charlier approximations, controlled by Poisson-weighted averages of the absolute values of higher-order forward differences of the coefficient sequence. This complements classical analytic depoissonization, which instead relies on complex-plane growth estimates: classical methods are natural when such estimates are available, whereas the present approach is useful when the relevant finite differences can be bounded, without requiring estimates at nonreal arguments. We also study the inverse problem of deriving asymptotic expansions of the Poisson transform at large positive arguments from the coefficient sequence. This leads to finite-difference analogues of Ramanujan's expansion. Applications to combinatorics and probability are presented.

math.CO↗
arXiv2021

Poisson approximation in $χ^2$ distance by Chen-Stein approach

The main purpose of the paper is to investigate the possibility of applying Chen-Stein approach to estimate the $χ^2$ distance between Poisson distribution and a sum of independent indicators. Earlier results concerning $χ^2$ distance between above mentioned distributions either used analytical approach heavily based on the analysis of the generating functions or on rather lengthy and complicated elementary calculations. Applying Chen-Stein approach we succeed in providing a very quick proof of upper bounds for $χ^2$ distance that are of comparable strength to the earlier estimates obtained by other approaches.

math.PR↗
arXiv2014

Distribution of the sum-of-digits function of random integers: a survey

We review some probabilistic properties of the sum-of-digits function of random integers. New asymptotic approximations to the total variation distance and its refinements are also derived. Four different approaches are used: a classical probability approach, Stein's method, an analytic approach and a new approach based on Krawtchouk polynomials and the Parseval identity. We also extend the study to a simple, general numeration system for which similar approximation theorems are derived.

math.PR↗
arXiv2013

An analytic approach to the asymptotic variance of trie statistics and related structures

We develop analytic tools for the asymptotics of general trie statistics, which are particularly advantageous for clarifying the asymptotic variance. Many concrete examples are discussed for which new Fourier expansions are given. The tools are also useful for other splitting processes with an underlying binomial distribution. We specially highlight Philippe Flajolet's contribution in the analysis of these random structures.

math.CO↗
arXiv2013

Limit laws of the coefficients of polynomials with only unit roots

We consider sequences of random variables whose probability generating functions are polynomials all of whose roots lie on the unit circle. The distribution of such random variables has only been sporadically studied in the literature. We show that the random variables are asymptotically normally distributed if and only if the fourth normalized (by the standard deviation) central moment tends to 3, in contrast to the common scenario for polynomials with only real roots for which a central limit theorem holds if and only if the variance goes unbounded. We also derive a representation theorem for all possible limit laws and apply our results to many concrete examples in the literature, ranging from combinatorial structures to numerical analysis, and from probability to analysis of algorithms.

math.PR↗
arXiv2012

Analysis of an exhaustive search algorithm in random graphs and the n^{c\log n} -asymptotics

We analyze the cost used by a naive exhaustive search algorithm for finding a maximum independent set in random graphs under the usual G_{n,p} -model where each possible edge appears independently with the same probability p. The expected cost turns out to be of the less common asymptotic order n^{c\log n}, which we explore from several different perspectives. Also we collect many instances where such an order appears, from algorithmics to analysis, from probability to algebra. The limiting distribution of the cost required by the algorithm under a purely idealized random model is proved to be normal. The approach we develop is of some generality and is amenable for other graph algorithms.

math.PR↗
arXiv2010

A Tauberian theorem for Ingham summation method

The aim of this work is to prove a Tauberian theorem for the Ingham summability method. The Tauberian theorem we prove is then applied to analyze asymptotics of mean values of multiplicative functions on natural numbers.

math.NT↗
arXiv2010

Asymptotic variance of random symmetric digital search trees

Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of the Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic de-Poissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising $n(\log n)^2$-variance for certain notions of total path-length is also clarified.

math.CO↗
arXiv2009

Distribution of Random Variables on the Symmetric Group

The well known Erdos-Turan law states that the logarithm of an order of a random permutation is asymptotically normally distributed. The aim of this work is to estimate convergence rate in this theorem and also to prove analogous result for distribution of the logarithm of an order of a random permutation on a certain class of subsets of the symmetric group. We also study the asymptotic behavior of the mean values of multiplicative functions on the symmetric group and the results we obtain are of independent interest besides their application to the investigation of the remainder term in the Erdos-Turan law. We also study a related problem of distribution of the degree of a splitting field of a random polynomial and obtain sharp estimates for its convergence rate to normal law. In research we apply both probabilistic and analytic methods. Some analytic methods used here have their origins in the probabilistic number theory, and some have their roots in the theory of summation of divergent series. One of the approaches we use is to apply Tauberian type estimates for Voronoi summability of divergent series to analyze the generating functions of the mean values of multiplicative functions.

math.CO↗
arXiv2008

A Charlier-Parseval approach to Poisson approximation and its applications

A new approach to Poisson approximation is proposed. The basic idea is very simple and based on properties of the Charlier polynomials and the Parseval identity. Such an approach quickly leads to new effective bounds for several Poisson approximation problems. A selected survey on diverse Poisson approximation results is also given.

math.PR↗