Abstract
where f (x) is any continuous monotonic function, and with the modification of given sequences in such a way as to make them satisfy the condition obtained. By the ilelly-Bray theorem,2 (2) is valid if (1) is a bounded sequence. It will be shown that (2) remains valid if (1) is unbounded but does not increase too rapidly in a sense to be specified below (Definition (4), Theorem I). Such a sequence will be called normal. It will also be shown that any sequence with an asymptotic distribution can be normalized by a simple modification which does not alter its distribution function. In the case of a continuous distribution function +(x) the modification consists in the suppression of a subsequence, the number of terms suppressed from x1, * *, x being o(n) (Theorem II). This is a best possible result as, conversely, the
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