I have a following setup: Let $c\in{\Bbb R}$, $R^2\in [0,1]$ and $\Psi,\varepsilon_1,\varepsilon_2,\ldots$ independent random variables on a probability space $(\Omega,{\cal A},{\Bbb P})$. Define the ability-to-pay variables by \begin{equation} A_j(R^2):=\sqrt{R^{2}}\Psi+\sqrt{1-R^{2}}\varepsilon_j \end{equation} for $j\in{\Bbb N}$. The unconditional and conditional default probabilities are respectively defined by \begin{eqnarray} p(R^2) & := & {\Bbb P}(A_j\leq c)={\Bbb P}(\sqrt{R^{2}}\Psi+\sqrt{1-R^{2}}\varepsilon_j\leq c), \\ p(\psi,R^2) & := & {\Bbb P}(\sqrt{R^{2}}\psi+\sqrt{1-R^{2}}\varepsilon_j\leq c) \end{eqnarray} for $\Psi\in{\Bbb R}$ and any $j\in{\Bbb N}$. Default indicators and default counts are defined by $$ I_j(R^2):={\bf 1}_{\{A_j(R^2)\leq c\}}$$ and $$D_n(R^2):=\sum_{j=1}^n I_j(R^2)$$ for $n\in{\Bbb N}$.
Let $X$ be a random variable and assume that the random variables $\Psi,X $ are standardized and follow a multivariate normal distribution. It is known that conditional on $\Psi$, $\frac{D_n(R^2)}{n}$ coverges in probability to $p(\Psi,R^2)$
I want to understand how to show following: $$\lim_{n\rightarrow\infty}\rho_{\tau}(X,D_n(R^2))=\rho_{\tau}(X,p(\Psi,R^2))$$ where $\rho_{\tau}$ denotes Kendall's Tau.
Thank you in advance. Any ideas on the proof are very welcome, as well as the assesment of its complexity :)
Best, Dima