On the Lower Limit of Sums of Independent Random Variables

1. Let X1 , X2 , • • • , Xn . . . be independent random variables and let Sn = X, . In the so-called law of the iterated logarithm, completely solved by Feller recently, the upper limit of S n as n -4 co is considered and its true order of magnitude is found with probability one . A counterpart to that problem is to consider the lower limit of Sn as n --> oo and to make a statement about its order of magnitude with probability one . THEOREM 1 . Let XI , • • •, Xn , • • be independent random variables with the common distribution : Pr(X,b = 1) = p, Pr(X,,, = 0) = 1 p = q. Let ¢(n) I oo and