Let X1, X2,... be, i.i.d. random variables, and put \( S_{n}=X_{1}+\cdots+X_{n}\). We find necessary and sufficient moment conditions for \(\int_{\varepsilon }^{\infty }f(x^{q})dx < \infty , \varepsilon >\delta \), where δ≥ 0 and q>0, and \(f(x)=\sum_{n}a_{n}P(\left\vert S_{n}\right\vert >xb_{n})\) with a n >0 and b n is either \(n^{1/p},\,0<p<2,\,\sqrt{n\,\log\,n}\) or \(\sqrt{n\,\log\,\log\,n}.\) The series f(x) we deal with are classical series studied by Hsu and Robbins, Erdős, Spitzer, Baum and Katz, Davis, Lai, Gut, etc
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Li, D., Spătaru, A. Refinement of Convergence Rates for Tail Probabilities. J Theor Probab 18, 933–947 (2005). https://doi.org/10.1007/s10959-005-7534-2
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DOI: https://doi.org/10.1007/s10959-005-7534-2