Hazut Netanel, Medalion Shlomi, Kessler David A, Barkai Eli
Department of Physics and Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 May;91(5):052124. doi: 10.1103/PhysRevE.91.052124. Epub 2015 May 14.
In this article we generalize the classical Edgeworth expansion for the probability density function (PDF) of sums of a finite number of symmetric independent identically distributed random variables with a finite variance to PDFs with a diverging variance, which converge to a Lévy α-stable density function. Our correction may be written by means of a series of fractional derivatives of the Lévy and the conjugate Lévy PDFs. This series expansion is general and applies also to the Gaussian regime. To describe the terms in the series expansion, we introduce a new family of special functions and briefly discuss their properties. We implement our generalization to the distribution of the momentum for atoms undergoing Sisyphus cooling, and show the improvement of our leading order approximation compared to previous approximations. In vicinity of the transition between Lévy and Gauss behaviors, convergence to asymptotic results slows down.
在本文中，我们将经典的埃奇沃思展开推广到具有发散方差的概率密度函数（PDF），该函数适用于有限数量具有有限方差的对称独立同分布随机变量之和，其收敛于一个 Lévy α - 稳定密度函数。我们的修正可以通过 Lévy 和共轭 Lévy PDF 的一系列分数阶导数来表示。这个级数展开是通用的，也适用于高斯区域。为了描述级数展开中的项，我们引入了一个新的特殊函数族，并简要讨论了它们的性质。我们将我们的推广应用于经历西西弗斯冷却的原子的动量分布，并展示了与先前近似相比我们的主导阶近似的改进。在 Lévy 和高斯行为之间的转变附近，向渐近结果的收敛会减慢。
