A Central Limit Theorem for Some Special Complex-Valued Probability Densities

Abstract

We describe the behavior of the n-fold convolutions[formula] under a suitable scaling as n→∞, where f is an integrable complex-valued function on R. We consider only the unstable case and only two typical examples in that case. In the first example f is defined by f̂(t)=exp{i|t|p−|t|q} and in the second one by f̂(t)=exp{i|t|psgn(t)−|t|q}, where p and q are real numbers such that 2≤p<q.