Before revision:  We derive the asymptotic distributions of these test statistics and their permutation distributions under randomness, under a set of general stationary processes, and under a Markov model of "streakiness" , which allow us to evaluate their local asymptotic power.
After revision:  We derive the asymptotic distributions of these test statistics and their permutation distributions under randomness, under a set of general stationary processes, and under a class of Markov chain alternatives , which allow us to evaluate their local asymptotic power.