The main purpose of the paper is to give necessary and sufficient conditions for the almost sure boundedness of (S n -a r )/B(n), where S n = A", +X 2 + ... + X n , A' r being independent and identically distributed random variables, and %" and B(n) being centering and norming constants. The conditions take the form of the convergence or divergence of a series of a geometric subsequence of the sequence P{S n -y. n > a B{n)), where a is a constant. The theorem is distinguished from previous

more »

... ar results by the comparative weakness of the subsidiary conditions and the simplicity of the calculations. As an application, a law of the iterated logarithm general enough to include a result of Feller is derived.