A Weak Law of Large Numbers for Empirical Measures via Stein's Method

Abstract

Let $E$ be a locally compact Hausdorff space with countable basis and let $(X_i)_{\i\in\mathbb{N}}$ be a family of random elements on $E$ with $(1/n) \sum^n_{i=1} \mathscr{L}(X_i) \Rightarrow^v \mu (n \rightarrow \infty)$ for a measure $\mu$ with $\|\mu\| \leq 1$. Conditions are derived under which $\mathscr{L} ((1/n) \sum^n_{i=1} \delta_{Xi}) \Rightarrow^w \delta_\mu(n \rightarrow \infty)$, where $\delta_x$ denotes the Dirac measure at $x$. The proof being based on Stein's method, there are generalisastions that allow for weak dependence between the $X_i$'s. As examples, a dissociated family and an immigration-death process are considered. The latter illustrates the possible applications in proving convergence of stochastic processes.