Upcrossing Inequalities for Stationary Sequences and Applications

Let g be a function which assigns to each stationary process (Xn) ∞ n=1 and to each sample X1 . . . Xn of the process a real number g(X1, . . . , Xn), which may also depend on the distribution of (Xn). We obtain effective bounds on the probability that the sequence (g(X1, . . . , Xn)) ∞ n=1 crosses a fixed interval some number of times in terms of a quantity measuring the “average sub-additivity” of g. As applications we derive universal upcrossing inequalities for Kingman’s sub-additive ergodic theorem, the ShannonMcMillan-Breiman theorem and the Kolmogorov complexity statistic.