 A more rigorous treatment of testing the random walk null hypothesis for United States stock prices using one-sided optimal statistical tests is due to Alok Bhargava.It is important to separately test the random walk null hypothesis against stationary and explosive alternatives using two one-sided tests.In contrast, two-sided asymptotic tests such as those employed by Lo and MacKinlay can lead to misleading conclusions because the process could be stationary or explosive under the alternative hypothesis.