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

A strong ergodic theorem for extreme and intermediate order statistics