Consider the following equations:
$$ \widehat{R}(f) = \frac{1}{n} \sum_{i=1}^n (Y_i - f(X_i))^2$$
$$ R(f) = \mathbb{E} (Y - f(X))^2$$
Suppose that we have $\min_{f \in \mathcal{F}} \widehat{R}(f) \stackrel{p}{\rightarrow} \min_{f \in \mathcal{F}} R(f)$.
Let $\widehat{f}$ be the minimizer of $\widehat{R}(f)$ and $f^*$ the minimizer of $R(f)$. Then is it true that we have $\widehat{f} \stackrel{p}{\rightarrow} f^*$ in some sense?
E.g., perhaps something like $\int | \widehat{f}(x) - f^*(x)| p(x) dx \stackrel{p}{\rightarrow} 0$?