  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.
