随机变量的倒谱分析：粘液剂
Cepstral Analysis of Random Variables: Muculants
摘要：
提出了一种用于描述离散随机变量的参数化描述方法，称为粘粒。与累积量相反，粘液量基于特征函数的对数的傅里叶级数展开，而不是泰勒级数展开。我们利用倒谱理论的结果，推导出基本性质的粘液，其中一些表现出优于那些累积量的行为。例如，粘液量和累积量都是加性的。虽然累积量的存在与特征函数可微的频率有关，但如果特征函数满足Paley-Wiener条件，则所有的累积量都存在。此外，粘液序列，如果随机变量有有限的期望，从它的粘液的特征函数的重建收敛。我们进一步发展了粘粒和累积量之间的联系，并提出了选定的离散随机变量的粘粒。具体来说，它示出的泊松分布是唯一的分布，只有前两个粘液是非零的。
英文摘要：
An alternative parametric description for discrete random variables, called muculants, is proposed. In contrast to cumulants, muculants are based on the Fourier series expansion, rather than on the Taylor series expansion, of the logarithm of the characteristic function. We utilize results from cepstral theory to derive elementary properties of muculants, some of which demonstrate behavior superior to those of cumulants. For example, muculants and cumulants are both additive. While the existence of cumulants is linked to how often the characteristic function is differentiable, all muculants exist if the characteristic function satisfies a Paley-Wiener condition. Moreover, the muculant sequence and, if the random variable has finite expectation, the reconstruction of the characteristic function from its muculants converge. We furthermore develop a connection between muculants and cumulants and present the muculants of selected discrete random variables. Specifically, it is shown that the Poisson distribution is the only distribution where only the first two muculants are nonzero.
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