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.
Citation
Gesine Reinert. "A Weak Law of Large Numbers for Empirical Measures via Stein's Method." Ann. Probab. 23 (1) 334 - 354, January, 1995. https://doi.org/10.1214/aop/1176988389
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