On some statistical tests for randomness

This paper presents statistical tests for randomness. We consider the problems of testing whether or not a strictly stationary sequence is an independent sequence and a given permutation is a random permutation. We construct the test based upon an estimator of a "distance to whiteness" introduced by Drouiche (1994) and the test based on the Tracy-Widom distribution of the length of a longest monotonically increasing subsequence of a permutation. Asymptotic distributions of the test statistics are found. Numerical experiments are carried out to emphasize the performance of the tests