Bulletin of the
Korean Mathematical Society BKMS

The rate of convergence for an almost surely convergent series $S_n=\sum_{i=1}^n X_i$ of independent random variables is studied in this paper. More specifically, when $S_n$ converges almost surely to a random variable $S$, the tail series $T_n\equiv S-S_{n-1}=\sum_{i=n}^\infty X_i$ is a well-defined sequence of random variables with $T_n\to 0$ almost surely. Conditions are provided so that for a given positive sequence $\{b_n, n\ge 1\}$, the limit law $\sup_{k\ge n}|T_k|/b_n \overset P \to \rightarrow 0$ holds. This result generalizes a result of Nam and Rosalsky [4].

Keywords: convergence in probability, tail series, independent random variables, weak law of large numbers, almost sure convergence

MSC numbers: 60F05, 60F15