Short Communications

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

Gy. Michaletzky^a, L. Szeidl^a, P. Várlaki<sup>b

a</sup> Department of Telecommunication and Telematics, Technical University, Hungary
^b Department of Probability Theory and Statistics, Eötvös Loránd University, Hungary

Abstract: 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 $0 < \alpha < 2$, $\alpha\ne 1$, and that of Fresnel functions $S$ and $C$ for $\alpha=1$.

Keywords: normalized power sums, domain of attraction of $\alpha$-stable law, limit characteristic function, generalized hypergeometric functions, Fresnel $S$ and $C$ functions, Schrö, dinger equation.

Received: 10.06.1997

Language: English

DOI: 10.4213/tvp937

 English version: Theory of Probability and its Applications, 1999, 43:1, 126–134

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