Asymptotic results for random sums of dependent random variables

Our main result is a central limit theorem for random sums of the form ∑i=1NnXi, where {Xi}i≥1{Xi}i≥1 is a stationary mm-dependent process and NnNn is a random index independent of {Xi}i≥1{Xi}i≥1. This extends the work of Chen and Shao on the i.i.d. case to a dependent setting and provides a variation of a recent result of Shang on mm-dependent sequences. Further, a weak law of large numbers is proven for ∑i=1NnXi, and the results are exemplified with applications on moving average and descent processes.