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Abstract

We demonstrate the relevance of the Prokhorov inequality to the study of Khintchine-type inequalities in symmetric function spaces. Our main result shows that the latter inequalities hold for a pair of quasi-Banach symmetric function spaces X and Y if and only if the Kruglov operator K acts from X to Y . We also obtain an extension of von Bahr-Esseen and Esseen-Janson L p -estimates for sums of independent mean zero random variables to the class of quasi-Banach symmetric spaces. In particular, in contrast to the well-known Esseen-Janson theorem, we do not assume that the summands are equidistributed.