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

Let E be a locally compact Hausdorff space with countable basis and let (X i ) i ∈ N be a family of random elements on E with (1/n) Σ i=1 n L(X i ) ⇒ H(n → ∞) for a measure μ with ∥μ∥ ≤ 1. Conditions are derived under which L((1/n)Σ i=1 n δx i ) ⇒ δ μ (n→∞), where δ x denotes the Dirac measure at x. The proof being based on Stein's method, there are generalisations 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.