Convergence rate of precise asymptotics in the Baum-Katz laws of large numbers

Abstract

Let $\{X, X_n, n\ge 1\}$ be a sequence of independent and identically distributed (i.i.d.) random variables and $S_n=X_1+X_2+\cdots+X_n$. In the present paper, we study the precise asymptotics for the following series
$$
\sum_{n=1}^\infty\pp\left(\left|S_n\right|\ge \varepsilon n^{1/p}\right)\ \ \ \text{for all}\ \ \varepsilon>0,
$$
where $1\le p< 2$, and consider the convergence rate of the series, which extends the works in He and Xie \cite{He-Xie}.