Riesz fractional integrals and complex fractional moments for the probabilistic characterization of random variables

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• 作者：Mario Di Paola ; ^{mario.dipaola@unipa.it} ; Francesco Paolo Pinnola ^{francescopaolopinnola@gmail.com}
• 关键词：Fractional calculus ; Mellin transform ; Complex order moments ; Fractional moments ; Fractional spectral moments ; Probability density function ; Characteristic function
• 刊名：Probabilistic Engineering Mechanics
• 出版年：2012
• 期刊代码：146_02668920
• 类别：et
• 出版时间：July, 2012
• 卷：29
• 期：Complete
• 页码：149-156
• 文件大小：387 K

摘要

The aim of this paper is the probabilistic representation of the probability density function (PDF) or the characteristic function (CF) in terms of fractional moments of complex order. It is shown that such complex moments are related to Riesz and complementary Riesz integrals at the origin. By invoking the inverse Mellin transform theorem, the PDF or the CF is exactly evaluated in integral form in terms of complex fractional moments. Discretization leads to the conclusion that with few fractional moments the whole PDF or CF may be restored.

Application to the pathological case of an -stable random variable is discussed in detail, showing the impressive capability to characterize random variables in terms of fractional moments.