Comments on the paper: A bilateral inequality on the Borel–Cantelli Lemma

A new extension of the Borel–Cantelli Lemma was obtained by Xie [Xie, Y.Q., 2008. A bilateral inequality on the Borel–Cantelli Lemma. Statist. Probab. Lett. 78, 2052–2057], and an important bilateral inequality on the Borel–Cantelli Lemma was given. However, there are several mistakes in the proof of Theorem 1 and Example 2 in the above cited reference. In this note we will present a corrected version of some results given therein.