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I wonder whether there might be a more general version of the Continuous Mapping Theorem (CMT), at least under certain conditions. What we know from the basic CMT is that $$ X_n \rightarrow X \text{ and } g \text{ continuous} \Rightarrow g(X_n) \rightarrow g(X),$$ where convergence can mean convergence of a real sequence, convergence in distribution, in probability and so on.

Now what if instead of a single function $g$ I have a sequence $(g_n)$ of continuous functions that converges (probably uniformly) to a continuous function $g$. Are there any conditions so that I can also conclude that $$ g_n(X_n) \rightarrow g(X)?$$

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    $\begingroup$ The standard approach is to look at the following identities and check what can be deduced from them: $$g_n(X_n) - g(X) = g_n(X_n) - g_n(X) + g_n(X) - g(X)$$ and $$g_n(X_n) - g(X) = g_n(X_n) - g(X_n) + g(X_n) - g(X)$$ -- the triangle inequality is your friend, here. $\endgroup$
    – Thomas
    Commented Apr 20, 2020 at 18:24

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Have a look at Theorem 1.11.1 in Van Der Vaart and Wellner, https://link.springer.com/book/10.1007/978-1-4757-2545-2: Copied here:

1.11.1 Theorem (Extended continuous mapping). Let $\mathbb{D}_{n} \subset \mathbb{D}$ and $g_{n}: \mathbb{D}_{n} \mapsto \mathbb{E}$ satisfy the following statements: if $x_{n} \rightarrow x$ with $x_{n} \in \mathbb{D}_{n}$ for every $n$ and $x \in \mathbb{D}_{0}$, then $g_{n}\left(x_{n}\right) \rightarrow g(x)$, where $\mathbb{D}_{0} \subset \mathbb{D}$ and $g: \mathbb{D}_{0} \mapsto \mathbb{E}$. Let $X_{n}$ be maps with values in $\mathbb{D}_{n}$, let $X$ be Borel measurable and separable, and take values in $\mathbb{D}_{0}$. Then (i) $X_{n} \leadsto X$ implies that $g_{n}\left(X_{n}\right) \leadsto g(X)$; (ii) $X_{n} \stackrel{\text { P* }}{\rightarrow} X$ implies that $g_{n}\left(X_{n}\right) \stackrel{\text { P* }}{\rightarrow} g(X)$; (iii) $X_{n} \stackrel{\text { as: }}{\rightarrow} X$ implies that $g_{n}\left(X_{n}\right) \stackrel{\text { as* }}{\rightarrow} g(X)$.

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