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.
Publié le : 1995-01-14
Classification:
Weak law of large numbers,
empirical measures,
Stein's method,
dissociated family,
immigration-death process,
60F05,
60G57,
60K25,
62G30
@article{1176988389,
author = {Reinert, Gesine},
title = {A Weak Law of Large Numbers for Empirical Measures via Stein's Method},
journal = {Ann. Probab.},
volume = {23},
number = {3},
year = {1995},
pages = { 334-354},
language = {en},
url = {http://dml.mathdoc.fr/item/1176988389}
}
Reinert, Gesine. A Weak Law of Large Numbers for Empirical Measures via Stein's Method. Ann. Probab., Tome 23 (1995) no. 3, pp. 334-354. http://gdmltest.u-ga.fr/item/1176988389/