The asymptotic behavior over a small parameter of a series of large deviation probabilities weighted with the Dirichlet divisors function
type: Item
instance of: scholarly article
title: The asymptotic behavior over a small parameter of a series of large deviation probabilities weighted with the Dirichlet divisors function
zbMATH Open document ID: 1116.60009
author: K.-H. Indlekofer
author: O. I. Klesov
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2007-03-20T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1229442620
review text: Consider partial sums \(~S(m,n)=\sum^m_{i=1} \sum^n_{j=1}X(i,j)\) of an array of i.i.d. random variables. The authors want to study the asymptotics, as \(~\varepsilon\downarrow 0\), of the series \[ \mathop{\sum}\limits^\infty_{m=1}\mathop{\sum}\limits^\infty_{n=1}{1\over{mn}}P(| S(m,n)|\geq mn\varepsilon). \] Making use of the asymptotics of Dirichlet divisors in number theory, this problem can equivalently be solved by deriving the asymptotics of the series \(\sum^\infty_{k=1} {{d_k}\over k} P(| S_k| \geq k\varepsilon),\) where \(S_k\) is the \(k\)th partial sum of i.i.d. summands having the same distribution as \(~X(1,1)\), and \(w_k={{d_k}\over k}\) are specific weights. The latter problem differs from the one-dimensional case, since the weights \(~w_k~\) do not vary regularly. Nevertheless, if the distribution function of the summands belongs to the domain of attraction to a stable law, the authors succeed in establishing the desired asymptotics. Key tool is a precise asymptotic for the cumulative sums of Dirichlet divisors.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 5135168
zbMATH Keywords: large deviation
zbMATH Keywords: complete convergence
zbMATH Keywords: stable law
zbMATH Keywords: domain of attraction
zbMATH Keywords: sums of independent random variables
zbMATH Keywords: random field
zbMATH Keywords: Dirichlet divisor
zbMATH Keywords: weighted sum of large deviation probabilities
reviewed by: J. G. Steinebach
MaRDI profile type: Publication
OpenAlex ID: W2049474338
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DOI: 10.7169/FACM/1229442620
description: scientific article; zbMATH DE number 5135168