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I was stuck the following problem: Let $X_n$, $X$, $n\ge 1$, be random variables defined on probability space $(\Omega,\mathcal{F},P)$. Prove that if $X_n\to X$ in probability then ${P_{{X_n}}} \Rightarrow {P_X}$ (weakly convergence, that is $\int {fd{P_{{X_n}}}} \to \int {fd{P_X}} $ for all $f\in C_b(\mathbb{R})$). Here ${P_X} = P \circ {X^{ - 1}}$, induced probability measure by $X$. Can someone help me?

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  • $\begingroup$ What is $C_b(\mathbb{R})$? $\endgroup$ – sinbadh Jan 15 '16 at 17:46
  • $\begingroup$ $C_b(\mathbb{R})$ is the set of all bounded continuous functions on $\mathbb{R}$. $\endgroup$ – User3101 Jan 15 '16 at 23:02
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For every $f\in C_b(\mathbb{R})$ one has that also $f(X_n)\to f(X)$ in probability. Since $f$ is bounded, the sequence $(f(X_n))_{n\geqslant 1}$ is uniformly integrable and hence $f(X_n)\to f(X)$ in $L^1$. This ensures that $\mathrm{E}[f(X_n)]\to\mathrm{E}[f(X)]$.

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