Upcrossing Inequalities for Stationary Sequences and Applications to Entropy and Complexity

An empirical statistic for a class C of stationary processes is a function g which assigns to each process (Xn) ∈ C with distribution P and to each sample X1 . . . Xn of the process a real number gP (X1, . . . , Xn). We describe a condition on g which implies that the sequence (gP (X1 . . . Xn)) ∞ n=1 obeys a (universal) upcrossing inequality, that is, that the probability that this sequence fluctuates across some interval k times decays to zero with k. As applications we get upcrossing inequalities for the ergodic theorem (recovering known results), and get upcrossing inequalities for the Shannon-McMillan-Breiman theorem and for the Kolmogorov complexity statistic.