Information regularity and the central limit question
Richard C. Bradley · Rocky Mountain Journal of Mathematics · 1983
TWO strictly stationary sequences (X k , k = ..., -1, 0, 1, ... ) of random variables are constructed for which the "information regularity" condition is satisfied, the second moments are finite, and Var (X x + • • • + X n ) -> oo as n -+ oo, but the central limit theorem fails to hold.In the first, the mixing condition based on "maximal correlations" is also satisfied.In the second, the mixing rate for "information regularity" is permitted to be arbitrarily fast (but w-dependence is not permitted) and n~2 yfox(X l + • • • + X n ) is permitted to approach 0 arbitrarily slowly.In the first sequence there is partial attraction of (X x + • • • + X n ) to some mixtures of mean-zero normal distributions, and in the second there is partial attraction to all infinitely divisible laws.