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Establishing a one-to-one correspondence between the interval (0, 1] and the collection of all subsequences of a given sequence s n , Buck and Pollard proved that s n is (C , 1)-summable if almost all of subsequences are, but not conversely. In this article, we consider the analogous questions for the p-Cesàro matrices. We show, for example, that if p > 1 2 then a bounded sequence is C p -summable if and only if almost all of its subsequences are C p -summable.