The -stable distributions introduced by Lévy 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 -stable distributions. These sequences are generalizations of independent and identically distributed -stable distributions and have not been previously studied. Long-range dependent -stable distributions might arise in the description of anomalous processes in nonextensive statistical mechanics, cell biology, finance. The parameter controls dependence. If then they are classical independent and identically distributed with -stable Lévy distributions. In the present paper we establish basic properties of -stable distributions and generalize the result of Umarov et al. [Milan J. Math.76, 307 (2008)], where the particular case was considered, to the whole range of stability and nonextensivity parameters and , respectively. We also discuss possible further extensions of the results that we obtain and formulate some conjectures.
We acknowledge thoughtful remarks by R. Hersh, E. P. Borges, and S. M. D. Queiros. Financial support by the Fullbright Foundation, SI International and NIH Grant No. P20 GM067594 (USA Agencies), and CNPq and Faperj (Brazilian Agencies) are acknowledged as well.
I. INTRODUCTION II. PRELIMINARIES AND AUXILIARY RESULTS A. Basic operations of -algebra B. -generalization of the exponential and cyclic functions C. -Fourier transform for symmetric densities III. WEAK CONVERGENCE OF CORRELATED RANDOM VARIABLES IV. SYMMETRIC -STABLE DISTRIBUTIONS AND THEIR PROPERTIES V. SCALING LIMITS OF SUMS OF -STABLE DISTRIBUTIONS VI. SCALING RATE ANALYSIS VII. ON ADDITIVE AND MULTIPLICATIVE DUALITIES VIII. CLASSIFICATION OF -STABLE DISTRIBUTIONS AND SOME CONJECTURES