Before revision:  We derive the asymptotic distributions of these test statistics and their permutation distributions under randomness and under general models of streakiness, which allows 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 Markov model of "streakiness", which allow us to evaluate their local asymptotic power.