Abstract

In this work, we study the partial sums of independent and identically distributed random variables with the number of terms following a fractional Poisson (FP) distribution. The FP sum contains the Poisson and geometric summations as particular cases. We show that the weak limit of the FP summation, when properly normalized, is a mixture between the normal and Mittag-Leffler distributions, which we call by Normal-Mittag-Leffler (NML) law. A parameter estimation procedure for the NML distribution is developed and the associated asymptotic distribution is derived. Simulations are run to check the performance of the proposed estimators under finite samples. An empirical illustration on the daily log-returns of the Brazilian stock exchange index (IBOVESPA) shows that the NML distribution captures better the tails than some of its competitors. Related problems such as a mixed Poisson representation for the FP law and the weak convergence for the Conway-Maxwell-Poisson random sum are also addressed.

摘要

在这项工作中，我们研究了具有遵循分数泊松 (FP) 分布的项数的独立同分布随机变量的部分和。FP 和包含泊松和几何求和作为特殊情况。我们表明，当适当归一化时，FP 求和的弱极限是正态分布和 Mittag-Leffler 分布之间的混合，我们称之为 Normal-Mittag-Leffler (NML) 定律。开发了 NML 分布的参数估计程序并推导出相关的渐近分布。运行模拟以检查在有限样本下提出的估计器的性能。巴西证券交易所指数 (IBOVESPA) 每日对数回报的实证说明表明，NML 分布比其某些竞争对手更好地捕捉尾部。