The asymptotic behavior over a small parameter of a series of large deviation probabilities weighted with the Dirichlet divisors function
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.
weighted sum of large deviation probabilities
scientific article; zbMATH DE number 5135168