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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).

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  • $\begingroup$ Studying the pointwise behavior of probability densities is problematic because they are unique up to measure-zero sets. In particular, $f$ and $f'$ can be two distinct densities of the same continuous random variable $X$, with $f(x) \ne f'(x)$ for all $x \in A$, where $A$ is countable (or more generally a null set). $\endgroup$ – jII Feb 15 '19 at 18:50
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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.

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