The alpha-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 (q,alpha)-stable distributions. These sequences are generalizations of independent and identically distributed 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 independent and identically distributed with alpha-stable Lévy distributions. In the present paper we establish basic properties of (q,alpha)-stable distributions and generalize the result of Umarov et al. [Milan J. Math. 76, 307 (2008)], where the particular case alpha=2,q[1,3) was considered, to the whole range of stability and nonextensivity parameters alpha(0,2] and q[1,3), respectively. We also discuss possible further extensions of the results that we obtain and formulate some conjectures.

中文翻译：

---

q>1 的对称 α 稳定 Lévy 分布的推广

Lévy 引入的 alpha 稳定分布在概率理论研究及其各种应用中发挥着重要作用，例如在统计物理学、生命科学和经济学中。在本文中，我们研究了分布具有渐近幂律衰减的长程相关随机变量的序列，它们被称为 (q,alpha) 稳定分布。这些序列是独立同分布的 alpha 稳定分布的推广，以前没有研究过。长期依赖 (q,alpha) 稳定分布可能出现在非广延性统计力学、细胞生物学、金融学中的异常过程描述中。参数 q 控制依赖性。如果 q=1，则它们是经典独立的，且服从 alpha 稳定的 Lévy 分布同分布。在本文中，我们建立了 (q,alpha) 稳定分布的基本性质，并推广了 Umarov 等人的结果。[米兰J.数学。76, 307 (2008)]，其中考虑了特定情况下 alpha=2,q[1,3)，分别针对稳定性和非扩展性参数 alpha(0,2] 和 q[1,3) 的整个范围。我们还讨论了我们获得的结果的可能进一步扩展，并提出了一些猜想。