A bilateral inequality on the Borel–Cantelli Lemma

Let A1,A2,…A1,A2,… be a sequence of events satisfying the condition ∑i=1∞P(Ai)=∞, IAiIAi be the indicator function for the event AiAi, Sn=∑i=1nIAi, Tn=SnI(Sn>0)/ESnTn=SnI(Sn>0)/ESn; we have an important bilateral inequality as follows: supp>0,p≠1lim supn→∞(ETnp)11−p⩽P(lim supAn)⩽infp<0lim infn→∞(ETnp)11−p and several of the well-known results on the generalizations of the Borel–Cantelli Lemma are special cases of this result.