On the characteristic function of the joint limit distribution of the first- and second-order power sums

In this paper we investigate the joint limit characteristic function of normalized first and second-order power sums for a sequence of i.i.d. random variables with distributions belonging to the domain of attraction of an alpha-stable law for alpha < 2. An explicit form is presented in terms of generalized hypergeometric functions when 0 < alpha < 2, alpha not equal 1, and that of Fresnel functions S and C for alpha = 1.