对称$α$ -稳定Lévy分布对$q>1$的推广
Generalization of symmetric $α$-stable Lévy distributions for $q>1$
摘要：
由Lévy引入的$α$ -稳定分布在概率理论研究及其各种应用中起着重要作用，例如，统计物理学、生命科学和经济学。本文研究分布具有渐近幂律衰减的长程相依随机变量序列，称之为$(q,α)$ -稳定分布。这些序列是独立同分布的推广。$α$ -稳定分布，以前没有研究过。长程相关的$(q,α)$ -稳定分布可能出现在非广延统计力学、细胞生物学、金融学中的异常过程的描述中。参数$q$控制相关性。如果是$q=1$，则它们是经典i.i.d.。具有$α$ -稳定Lévy分布。本文建立了$(q,α)$ -稳定分布的基本性质，并将Umarov，Tsallis和Steinberg（2008）的结果（其中考虑了$α=2, q\in [1,3),$的特殊情况）分别推广到稳定性和非扩展性参数$α\in (0,2]$和$q \in [1,3),$的整个范围.我们还讨论了可能的进一步推广的结果，我们得到的，并制定一些示意图。
英文摘要：
The $α$-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 $(q,α)$-stable distributions. These sequences are generalizations of i.i.d. $α$-stable distributions, and have not been previously studied. Long-range dependent $(q,α)$-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 $α$-stable Lévy distributions. In the present paper we establish basic properties of $(q,α)$-stable distributions, and generalize the result of Umarov, Tsallis and Steinberg (2008), where the particular case $α=2, q\in [1,3),$ was considered, to the whole range of stability and nonextensivity parameters $α\in (0,2]$ and $q \in [1,3),$ respectively. We also discuss possible further extensions of the results that we obtain, and formulate some conjectures.
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