A central limit theorem for some special complex-valued probability densities (Q1580459)

Two special integrable complex-valued probability density functions, which are unstable in the sense that the logarithms of their Fourier transformations converge to zero faster than the \(q\)th (\(q\) greater than or equal to 2) power of \(t\) as \(t\) tends to 0, are considered. It is shown that, under a special scaling, the \(n\)-fold convolutions of these functions diverge in a very regular oscillatory manner. Some results on the asymptotic behavior of the \(L_1\) norms of these convolutions are also obtained as a corollary of the main theorem.