Compound Poisson process approximation for locally dependent real-valued random variables via a new coupling inequality

Abstract

We present a general and quite simple upper bound for the total variation distance $d_{\mathrm{TV}}$ between any stochastic process $(X_i{)}_{i\in \Gamma }$ defined over a countable space $\Gamma$, and a compound Poisson process on $\Gamma .$ This result is sufficient for proving weak convergence for any functional of the process $(X_i{)}_{i\in \Gamma }$ when the real-valued $X_i$ are rarely non-zero and locally dependent. Our result is established after introducing and employing a generalization of the basic coupling inequality. Finally, two simple examples of application are presented in order to illustrate the applicability of our results.