On the rate of decay of concentration functions of <Emphasis Type="Italic">n</Emphasis>-fold convolutions of probability distributions

Concentration functions of n-fold convolutions of probability distributions is shown to exhibit the following behavior. Let φ(n) be an arbitrary sequence tending to infinity as n tends to infinity, and ψ(x) be an arbitrary function tending to infinity as x tends to infinity. Then there exists a probability distribution F of a random variable X such that the mathematical expectation E _ψ(|X|) is infinite and, moreover, the upper limit of the sequence \(\sqrt n \phi \left( n \right)Q_n\) is equal to infinity, where Q _n is the maximal atom of the n-fold convolution of distribution F. Thus, no infinity conditions imposed on the moments can force the concentration functions of n-fold convolutions decay essentially faster than o(n ^?1/2).