Limit distribution of the maximum and minimum of successive cumulative sums of random variables

as iV—> 00. In the particular case when JUJV = 0 and a = 6, the limit values of (1.2) and (1.3) were recently obtained by Erdös and Kac [ l ] . 1 The case when [INT^O, especially when /i^iV converges to a finite value, is of particular importance in the theory of sequential tests of statistical hypotheses. I t will be seen in §3 that the limit distribution of the number of observations required by a sequential probability ratio test can immediately be obtained from the limit values of (1.2) and (1.3), and vice versa.