The asymptotic distribution of the range and other functions of partial sums of stationary processes

Let ℰ n , n = 1, 2, ⋯ , be the net input in a reservoir during the n th period of time, and set S 0 = 0, S n = ℰ 1 + … + ℰ n , = 1, 2, ⋯ . Many quantities of interest, such as range, first‐passage times, and duration of deficit period, are functions of the partial sums S n . In this paper it is pointed out that the functional central limit theorem, which has been previously used to obtain asymptotic results for independent and identically distributed ℰ n , can be applied to a class of stationary sequences as well. To this class belong m ‐dependent, Markov, autoregressive, and autoregressive‐moving average types of stationary processes.