On a theorem of buck and pollard

Abstract

A short proof is given for the following theorem proved first by Buck and Pollard: Let {S {s k} be a (C, 1) summable sequence, for which $$\sum\limits_{k = 1}^\infty {\frac{{S_k^2 }}{{k^2 }} < \infty holds} $$ holds. Then almost all subsequences of {S {s k} are (C, 1)-summable and the (C, 1)-limit of almost all subsequences is the same as that of the original sequence. Here “almost all∝ means the following: Let t be a real number with the dyadic expansion $$t = \sum\limits_{k = 1}^\infty {\varepsilon _k } (t) 2^{ - k} , \varepsilon _k (t) = 0$$ and consider, together with {s k} the sequence $$(*) \{ S_{\lambda _k (t)} \} (k = 1,2,...),$$ where λk(t) is the index of the k-th 1 in the sequence {ɛk}. Then all sequences (*) are (C, 1) summable, except for a set measure zero in t. The proof is based on the strong law of large numbers by Kolmogorov.