Statistical gap Tauberian theorems in metric spaces

• Published in: Journal of mathematical analysis and applications Vol. 282; no. 2; pp. 744 - 755
• Main Authors: Fridy, J.A., Khan, M.K.
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 15-06-2003; Elsevier
• Subjects: Central limit theorem; Circle methods; Convolution methods; Exact sciences and technology; Hausdorff methods; Limit theorems; Mathematics; Probability and statistics; Probability theory and stochastic processes; Random walk methods; Sciences and techniques of general use; Statistical convergence; Hausdorff methods; Random walk methods; Circle methods; Central limit theorem; Convolution methods; Statistical convergence; Statistical method; Statistical gap; Tauberian theorem; Random walk; Probability theory; Circle method; Convergence; Hausdorff method
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/S0022-247X(03)00248-8

By using the concept of statistical convergence we present statistical Tauberian theorems of gap type for the Cesàro, Euler–Borel family and the Hausdorff families applicable in arbitrary metric spaces. In contrast to the classical gap Tauberian theorems, we show that such theorems exist in the statistical sense for the convolution methods which include the Taylor and the Borel matrix methods. We further provide statistical analogs of the gap Tauberian theorems for the Hausdorff methods and provide an explanation as to how the Tauberian rates over the gaps may differ from those of the classical Tauberian theorems.