Borel-Cantelli引理的一个推论

ONE OF BOREL-CANTELLI LEMMA COROLLARY

文献[5]的结果是对概率论中一个重要的引理Borel-Cantelli引理的推广,文献[2]给出了更一般的双边Borel-Cantelli引理不等式,作者主要对此双边Borel-Cantelli引理不等式在p=2与p=-2的情形下做了进一步的推广,主要结果为A1,A2…是一列概率空间中的事件,满足∑∞k=1kαP(Ak)=∞(α≥0),那么有limn→∞sup(∑nk=1kαP(Ak))2∑nk=1∑ni=1kαiαP(AiAk)≤P(limsupAn)≤limn→∞inf[(∑nk=1kαP(Ak))2∑nk=1∑ni=1kαiαP(AiAk)]31.这一结果是对文献[5]的结论进一步的推广.

The result of literature [5] is very important in probability theory as a lemma Borel-Cantelli lemma,literature [2] gives a more general inequality about bilateral Borel-Cantelli lemma.This article is bilateral Borel-Cantelli lemma inequality with the case p=2 and p=-2 for further promotion,main results is that A1,A2… is a sequence of event in probability space and ∑∞k=1kαP（Ak）=∞（α≥0）,thenlimn→∞sup（∑nk=1kαP（Ak））2∑nk=1∑ni=1kαiαP（AiAk）≤P（limsupAn）≤limn→∞inf[（∑nk=1kαP（Ak））2∑nk=1∑ni=1kαiαP（AiAk）]13.Also is further promotion of the conclusions of literature [5].