A strong ergodic theorem for extreme and intermediate order statistics

We study almost sure limiting behavior of extreme and intermediate order statistics arising from strictly stationary sequences. First, we provide sufficient dependence conditions under which these order statistics converges almost surely to the left or right endpoint of the population support, as in the classical setup of sequences of independent and identically distributed random variables. Next, we derive a generalization of this result valid in the class of all strictly stationary sequences. For this purpose, we introduce notions of conditional left and right endpoints of the support of a random variable given a sigma-field, and present basic properties of these concepts. Using these new notions, we prove that extreme and intermediate order statistics from any discrete-time, strictly stationary process converges almost surely to some random variable. We describe the distribution of the limiting variate. Thus we establish a strong ergodic theorem for extreme and intermediate order statistics.