On the maximum partial sums of sequences of independent random variables

Two types of fundamental limit theorems are known about S,, the one clustering around the central limit theorem and the other the law of the iterated logarithm. In 1945 Feller 12(2) called attention to the study of the behavior of Sn*. Since then an important result has been obtained by Erd6s and Kac 8, namely, the limiting distribution of S * for sufficiently general sequences of Xn. This corresponds to the central limit theorem for Sn. Now under certain conditions when the distribution of Sn tends to the normal distribution, an estimate of the difference of the two distributions has been given by Liapounoff 17, Cramer 5, Berry 3 and Essen 9. Cramer 6 and Feller 10 have also obtained more precise estimates for this difference for certain domains of variation of S, which proved essential to the general form of the law of the iterated logarithm. It is therefore of interest to make the same kind of investigations regarding S*. The problem is more difficult, since we have as yet no standard tools as in the case of Sn. We shall prove in this direction, as consequences of a more general but less handy inequality (Lemma 7), two theorems corresponding to the two types of estimation mentioned above. In order to state them we introduce the following notations. Let E(X) denote the mathematical expectation of X. We shall assume that for each X, the first moment is zero, and the third absolute moment is finite. Thus we can write