Iffis a bijection and is a non-negative and bounded then the limit exists and is equal to , where the limit is over all integer intervals

Suppose that for allt. IfIis an interval such that|I|>Bthen any infinite orbit intersectsI. The sum of over the points inIlying in a given infinite orbit is bounded above byIand below by|I|-2B.

If

Iis large enough then the sum of for can be made arbitrarily close to the number of infinite orbits off; the singleton orbits don't contribute since for those orbits. Thus in the limit the average ofdfover an interval of consecutive integers must become arbitrarily close to the number of infinite orbits of the permutation.