On Khintchine's Estimate for Large Deviations

Abstract

The large deviation estimate, used in classical proofs of the law of the iterated logarithm for i.i.d. random variables, implies the random variables satisfy a condition more stringent than a finite variance. Thus it is impossible to prove the law of the iterated logarithm in its full strength (i.e. assuming only a finite second moment) by using such a deviation estimate in a "straightforward" manner.