Umarov Sabir, Tsallis Constantino, Gell-Mann Murray, Steinberg Stanly
J Math Phys. 2010 Mar;51(3):33502. doi: 10.1063/1.3305292. Epub 2010 Mar 3.
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.
由列维引入的α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，我们研究具有渐近幂律衰减分布的长程相关随机变量序列，这些序列被称为（q，α）稳定分布。这些序列是独立同分布的α稳定分布的推广，此前尚未被研究过。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、金融中的异常过程描述中。参数q控制相关性。如果q = 1，那么它们就是具有α稳定列维分布的经典独立同分布。在本文中，我们建立了（q，α）稳定分布的基本性质，并将乌马罗夫等人[《米兰数学杂志》76, 307 (2008)]的结果进行了推广，其中考虑了α = 2，q ∈ [1, 3)的特殊情况，分别推广到了稳定性参数α ∈ (0, 2]和非广延性参数q ∈ [1, 3)的整个范围。我们还讨论了我们所获得结果可能的进一步扩展，并提出了一些猜想。
