A note on the bilateral inequality for a sequence of random variables

Two bilateral inequalities based on the Borel-Cantelli lemma and a non-negative sequence of bounded random variables were respectively obtained by Xie (2008, 2009). However, we observe that the upper bounds in the above cited references are greater than or equal to 1, so the upper bounds of these bilateral inequalities always hold true. In this note, we will extend the lower bound results on the assumptions that the random variables are neither non-negative nor bounded, which could be considered as a version of the Borel-Cantelli lemma with a random weight sequence. As an application, we also discuss the example given in Xie (2008) and Hu et al. (2009), and the best result is easily obtained for this example by taking the appropriate weight sequence.