Generalization of symmetric $\alpha$-stable L\'evy distributions for $q>1$

TL;DR

This paper introduces and analyzes a new class of long-range dependent distributions called $(q,eta)$-stable distributions, extending classical $eta$-stable laws to incorporate dependence parameterized by $q$, with potential applications in physics and finance.

Contribution

It generalizes classical $eta$-stable distributions to include dependence via the parameter $q$, establishing their basic properties and extending previous results to a broader parameter range.

Findings

Defined $(q,eta)$-stable distributions with power law decay.

Extended previous results to all $eta o (0,2]$ and $q o [1,3)$.

Discussed potential applications and future research directions.

Abstract

The $\alpha$-stable distributions introduced by L\'evy play an important role in probabilistic theoretical studies and their various applications, e.g., in statistical physics, life sciences, and economics. In the present paper we study sequences of long-range dependent random variables whose distributions have asymptotic power law decay, and which are called $(q,\alpha)$-stable distributions. These sequences are generalizations of i.i.d. $\alpha$-stable distributions, and have not been previously studied. Long-range dependent $(q,\alpha)$-stable distributions might arise in the description of anomalous processes in nonextensive statistical mechanics, cell biology, finance. The parameter $q$ controls dependence. If $q=1$ then they are classical i.i.d. with $\alpha$-stable L\'evy distributions. In the present paper we establish basic properties of $(q,\alpha)$-stable distributions, and generalize the result of Umarov, Tsallis and Steinberg (2008), where the particular case $\alpha=2, q\in [1,3),$ was considered, to the whole range of stability and nonextensivity parameters $\alpha \in (0,2]$ and $q \in [1,3),$ respectively. We also discuss possible further extensions of the results that we obtain, and formulate some conjectures.