Given a "almost independent" finite sequence of $\mathbb{R}$-valued random variables $X_1,X_2,\ldots,X_N$ having a common probability distribution $\mu$ (where ''almost independent'' could mean for example $|\mathrm{Cov}(X_i,X_j)| \leq \varepsilon_N$, with $\varepsilon_N \to 0$ as $N \to \infty$), and consider the so-called empirical measure $$\rho_N := \frac{1}{N} \sum_{k=1}^N \delta_{X_k}.$$ I am expecting to have results of the type $\rho_N \rightharpoonup \rho$ (weak convergence of probability measures) for some $\rho \in \mathcal{P}(\mathbb{R})$, where $\mathcal{P}(\mathbb{R})$ represents the space of probability measures on $\mathbb R$. What confuses me is that by definition of the weak convergence, I need to show $$''\int f \mathrm{d}\rho_N \to \int f \mathrm{d}\rho''$$ for all $f \in C_{\mathrm{b}}\left(\mathcal{P}(\mathbb{R}),\mathbb{R}\right)$. But what does the preceding sentence mean exactly? If I view $\rho_N$ as a "random vaeriable'' belonging to the space $\mathcal{P}(\mathbb{R})$, then $\int f \mathrm{d}\rho_N$ is clearly random and not deterministic. Also, how does function in $C_{\mathrm{b}}\left(\mathcal{P}(\mathbb{R}),\mathbb{R}\right)$ look like? By Riesz representation, I know the dual of $\mathcal{P}(\mathbb{R})$ is contained in $C_{\mathrm{b}}\left(\mathcal{P}(\mathbb{R}),\mathbb{R}\right)$, but I guess the space $C_{\mathrm{b}}\left(\mathcal{P}(\mathbb{R}),\mathbb{R}\right)$ is strictly larger than the dual of $\mathcal{P}(\mathbb{R})$? I am sorry my understanding is a bit vague. Thank you guys very much for the help!
Precise meaning of weak convergence of empirical measure (related to Interacting Particle Systems)
functional-analysisprobability theory
Best Answer
Can you be more specific about what kind of convergence you are looking for? Or is your question exactly about how to define convergence in this situation?
In empirical process theory, there are a few notions of convergence. Glivenko-Cantelli gives a type of uniform convergence of the form $$\sup_{x \in \mathbb{R}} |F_n(x) - F(x)| \overset{\text{almost surely}}{\to} 0$$ where $F_n$ is the CDF of the empirical measure. Donsker gives a type of "convergence in distribution" of random walks to Brownian motion in Skorohod space.