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If $X_i$ is a sequence of $\mathbb{R}^m$-valued random variables that converges either in probability or almost surely to $X$ and if $f$ is some measurable function from $\mathbb{R}^m$ into $\mathbb{R}$, does it follow that $f(X_i)$ converges to $f(X)$ as well?

I know about the continuous mapping theorem. Is there something similar for arbitrary measurable functions?

New Question: As the answer below suggested, let $\mathcal{C}$ be the set of functions $f$ that satisfy $f(X_i)$ converges in probability to $f(X)$ when $X_i$ converges in probability to $X$. What are the minimal properties of such a set?

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How about $X_i=1/i$ with probability one and $f(x)=1_{\{x\neq 0\}}$. Then $X_i\to X=0$ almost surely, but $f(X_i)=1 \nrightarrow 0=f(X)$. – Stefan Hansen Nov 9 '12 at 8:58
Note that you cannot avoid talking about continuity: If your statement holds for all random Variables (esp. for the constant ones), then $f$ is continuous. – martini Nov 9 '12 at 9:17
up vote 3 down vote accepted

Let $\cal C_1$ the collection of measurable functions from $\Bbb R^m$ to $\Bbb R$ such that whenever $\{X_n\}$ is a sequence of random variables with values in $\Bbb R^m$ converging to $X$ almost surely, then $f(X_n)\to f(X)$ almost surely.

We define in the same way $\cal C_2$ replacing "almost everywhere" by "in probability".

Fix $(x_1,\dots,x_m)\in\Bbb R^n$, and $\{(x_1^{(k)},\dots,x_m^{(k)})\}_k$ an arbitrary sequence converging to $(x_1,\dots,x_m)$. Let $X=(x_1,\dots,x_m)$ and $X_k=(x_1^{(k)},\dots,x_m^{(k)})$ (constant random variables). Then $X_k\to X$ in probability and almost surely. If $f\in\cal C_i$, then $f$ is continuous at $(x_1,\dots,x_m)$.

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