The classical Borel-Cantelli Lemma is a standard tool for deciding when an infinite number of rare events occur with probability one.It is a classical subject in probability and has many applications in dynamical systems. We propose an extension of Borel-Cantelli Lemma to characterize the multiple occurrence of events on the same time scale. Our results imply multiple Logarithm Laws for recurrence and hitting times of systems that are exponentially mixing of all orders. As applications, we prove an analogue of the Law of Iterated Logarithm for heavy-tailed random variables.
