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

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Abstract
Let ℰn, n = 1, 2, ⋯ , be the net input in a reservoir during the nth period of time, and set S0 = 0, Sn = ℰ1 + … + 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 Sn. 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 ℰn, 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.