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Let $X,X_1,X_2,...$ be real valued random variables on the same probability space $(\Omega, \mathcal{F},\mathbb{P})$. Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function.We know that by the open mapping theorem, if $X_n$ converges in distribution to $X$ , then $f(X_n)$ also converges in distribution to $f(X)$. But my question is suppose $X_1,X_2,...$ are not defined on the same probability space i.e. $(\Omega, \mathcal{F},\mathbb{P})$, instead they are defined on different probability spaces say, $(\Omega_n, \mathcal{F_n},\mathbb{P_n})$ for $n=1,2,..$, and let $X$ be defined on $(\Omega, \mathcal{F},\mathbb{P})$.

Then can we prove the open mapping theorem involving convergence in distribution?

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The question whether $X_n\stackrel{d}{\to} X$ (or not) does not depend on underlying probability spaces but purely on the distributions of $X_n$ and $X$. That also explains why we speak of "convergence in distribution" in that situation.

Actually it should not be looked at as a convergence of random variables but as a convergence of probability distributions (i.e. probability measures).

The latter are off course often induced by random variables, but they can also be missed.


edit (a proof that leaves out random variables)

Let $P$ be a probability measure and let $(P_n)_n$ be a sequence of probability measures all defined on measurable space $(\mathbb R,\mathcal B(\mathbb R))$.

Let $f:\mathbb R\to\mathbb R$ be a continuous function.

Further let it be that: $$\lim_{n\to\infty}\int gdP_n=\int gdP\text{ for every continuous and bounded function }g:\mathbb R\to\mathbb R$$This actually states that $P_n$ converges weakly to $P$ and the statement that $X_n\stackrel{d}{\to} X$ is exactly the statement that the distribution of $X_n$ weakly converges to the distribution of $X$.

Evidently we have:$$g:\mathbb R\to\mathbb R\text{ is bounded and continuous}\implies g\circ f:\mathbb R\to\mathbb R\text{ is bounded and continuous}$$

Consequently for any bounded and continuous $g:\mathbb R\to\mathbb R$ we find:$$\lim_{n\to\infty}\int gd(P_nf^{-1})=\lim_{n\to\infty}\int g\circ fdP_n=\int g\circ fdP=\int gdPf^{-1}$$where $Pf^{-1}$ denotes the probability measure on $(\mathbb R,\mathcal B(\mathbb R))$ that is prescribed by: $$B\mapsto P(f^{-1}(B))$$

This shows that $P_nf^{-1}$ converges weakly to $Pf^{-1}$.

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  • $\begingroup$ I have added a proof that leaves out underlying probability spaces. $\endgroup$
    – drhab
    Jun 5 '20 at 7:58

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