Fractional Poisson random sum and its associated normal variance mixture

TL;DR

This paper investigates the limit distribution of sums of i.i.d. variables with a fractional Poisson count, revealing a new Normal-Mittag-Leffler mixture distribution, and develops estimation methods with real-world stock data application.

Contribution

It introduces the Normal-Mittag-Leffler distribution as a limit law for fractional Poisson sums and provides parameter estimation and empirical validation.

Findings

The weak limit of fractional Poisson sums is a Normal-Mittag-Leffler mixture.

The proposed estimators perform well in finite samples.

The NML distribution better captures tail behavior in stock returns.

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 performed 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.