An optimal bound on the tail distribution of the number of recurrences of an event in product spaces
Let X-1, X-2,... be independent random variables and a a positive real number. For the sake of illustration, suppose A is the event that Xi+1 +...+ X-j greater than or equal to a for some integers 0 less than or equal to i < j < infinity. For each k greater than or equal to 2 we upper-bound the probability that A occurs k or more times, i.e. that A occurs on k or more disjoint intervals, in terms