I have two sequences of real random variables, $X_n$ and $\mu_n$. I know that
- $\mu_n$ converges weakly (in distribution) to a law $\mathcal N(0,1)$
- For every $\eta \in \mathbb R$, the law of $X_n\mid\mu_n=\eta$ converges in distribution to a law $\mathcal N(\eta,1)$
Can I say that the sequence $X_n$ itself converges in distribution to a law $\mathcal N(0,2)$, which is what we get when $$X|\mu \sim \mathcal N(\mu,1) \qquad \mu\sim \mathcal N(0,1)?$$
For now, the only idea that come to my mind is to use the fact that $X_n$ converges in distribution to a r.v. with the same law of $X$ if and only if
$$\forall h: \mathbb R \to \mathbb R,\ h\in C^0_0(\mathbb R)\qquad \mathbb E[h(X_n)]\to \mathbb E[h(X)]$$
but then, one has
$$ \lim_n \mathbb E[h(X_n)] = \lim_n \int_\mathbb R \mathbb E[h(X_n)\mid\mu_n=\eta]dP^{\mu_n}(\eta)$$
and since both the function of $\eta$ and the measure depend on $n$ I don't know how to apply any convergence theorem. Have you got any idea?
Edit: What can be relevant, in this case, is the uniform convergence of the sequence of functions $$f_n(\eta) = \mathbb E[h(X_n)\mid\mu_n=\eta]$$ in fact, if we have that $$f(\eta):=\mathbb E[h(\mathcal N(\eta,1))]\qquad \|f_n-f\|_\infty \to 0$$ then, $$\lim_n \int_\mathbb R \bigg ( \mathbb E[h(X_n)\mid\mu_n=\eta]-f(\eta) \bigg )dP^{\mu_n}(\eta)\le \lim_n \int_\mathbb R \|f_n-f\|_\infty dP^{\mu_n}(\eta)=0$$ Note that, since $f_n$ is uniformly bounded by $\sup h$ and converges pointwise to $f$ by assumption, it is sufficient to prove equicontinuity of $f_n$ and apply Ascoli-Atzerlà theorem.
Still, the general question is opened.
Here is an idea. Ideally we could want the following to hold: $$\begin{aligned}\lim_{n \to \infty}F_{X_n}(a)&=\lim_{n \to \infty}\int_\mathbb{R}\mathbb{I}_{(-\infty,a]}(x)f_{X_n}(x)dx=\\&=\int_\mathbb{R}\mathbb{I}_{(-\infty,a]}(x)\cdot \lim_{n \to \infty}\bigg(\int_\mathbb{R}f_{X_n|\mu_n}(x,y)f_{\mu_n}(y)dy\bigg)dx=\\ &=\int_\mathbb{R}\mathbb{I}_{(-\infty,a]}(x)\bigg(\int_\mathbb{R}f_{X|\mu}(x,y)f_{\mu}(y)dy\bigg)dx=\\ &=\Phi\bigg(\frac{a}{\sqrt{2}}\bigg),\, a \in \mathbb{R}\end{aligned}$$ where $\Phi$ is the standard normal cdf. So (ideally) if you could make some assumptions on the densities of the sequences, you could use convergence theorems and obtain the above.