Abstract
In this paper, we firstly introduce the definitions of upper semi-continuous function-valued (denoted by u. s. c. f. v., for short) stationary processes, and then we prove their properties. We mainly prove an ergodic theorem of u. s. c. f. v. stationary sequences in d-dimensional Euclidean space by using ”sandwich” method and famous Shapley-Folkman inequality, and the convergence is in the sense of generalized uniform Hausdorff metric.
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