Fix $n$, and consider random variables $x_1, \dots, x_n$ whose joint p.d.f. is $p_n$.
Assume that the empirical distribution of $x_1, \dots,x_n$ converges weakly almost surely to the probability density function function $\rho(x)$. Moreover, suppose that $x$'s are bounded.
Let $\hat{\rho}_n (y) = \frac{1}{n} \sum_{i =1}^n \delta(y - x_i)$, which is a p.m.f., and is random itself. Then,
$$ \hat{\rho}_n (y) \to \rho(y) \hspace{5pt} \text{weakly, almost surely under $p_n$} $$
Define the Hilbert transform of $\rho$ as $H[\rho](z) = {\rm PV} \frac{1}{\pi} \int \frac{\rho(x)}{z-x} \, dx$.
How can I show that $\lim_{n \to \infty} \mathbb{E}\Big[ \frac{1}{n} \sum_{i=1}^n H[\rho](x_i) \Big] = \int H[\rho](x) \rho(x) \, dx$? where the expectation is w.r.t. to $p_n$.