Are there any theorems that connect these two concepts, in particular, is there a result that states that convergence in probability of a sequence of continuous random variables $\{X_n\}_{n\geq 0}$ to another r.v. $X$ (also with a density) implies that the corresponding sequence of densities converge pointwise to the density of $X$? Or is there perhaps an obvious counterexample to this idea (c.f. convergence in distribution does not imply pointwise convergence of densities).
Probability Theory – Convergence in Probability and Pointwise Convergence of Densities
probabilityprobability distributionsprobability theorystatistics
Best Answer
You already know that convergence in distribution does not imply pointwise convergence of densities, i.e. there is a sequence of random variables $X_n \Rightarrow X$ such that the densities $f_n$ of $X_n$ do not converge pointwise to the density $f$ of $X$. But by the Skorohod representation theorem, there exists a probability space $\Omega$ and random variables $X_n', X'$ on $\Omega$ such that $X_n' \overset{d}{=} X_n$, $X' \overset{d}{=} X$, and $X_n' \to X'$ almost surely. In particular $X_n' \to X'$ in probability, and their densities are $f_n, f$ respectively, so the pointwise convergence of densities still fails.