Asymptotic Normality of Random Sums of m-dependent Random Variables

We prove a central limit theorem for random sums of the form ${\sum }_{i=1}^{N_n}{X}_i$, where $\{X_i{\}}_{i\ge 1}$ is a stationary $m-$dependent process and $N_n$ is a random index independent of $\{X_i{\}}_{i\ge 1}$. Our proof is a generalization of Chen and Shao’s result for i.i.d. case and consequently we recover their result. Also a variation of a recent result of Shang on $m-$dependent sequences is obtained as a corollary. Examples on moving averages and descent processes are provided, and possible applications on non-parametric statistics are discussed.