Convergence rate for a large deviation probability

David G. Kostka
1974 Proceedings of the American Mathematical Society  
It is known that a condition more stringent than a finite variance is needed to show, by a direct application of large deviation estimates, the convergence of .the series used in classical proofs of the law of the iterated logarithm. However, the series still converges if the variables have only a finite variance.
doi:10.1090/s0002-9939-1974-0345174-0 fatcat:r6ursrvpfvannidevcsnw3v77i