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Assume $X_n$ Converges in distribution to $X$ and $a_n \rightarrow a$ ,$b_n \rightarrow b$ where $a_n$ and $b_n$ are deterministic sequence,

prove $a_nX_n+b_n \rightarrow aX+b$ using characteristic function?

I have started this question as the following but I am not sure

$$ \phi_{a_nX_n+b_n}(t) = E\{e^{it(a_nX_n+b_n)}\} = e^{itb_n} E\{e^{ita_nX_n}\} \rightarrow e^{itb} E\{e^{itaX}\} = \phi_{aX+b}(t) $$

how can I show that $a_nX_n$ converges in distribution to $aX$?

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marked as duplicate by Davide Giraudo, Sasha, Michael Hoppe, Dilip Sarwate, Daniel Fischer Feb 26 '14 at 14:53

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  • $\begingroup$ Two random variables converge in distribution if their characteristic functions converge pointwise, and the pointwise limit is continuous at the origin. $\endgroup$ – fgp Feb 25 '14 at 17:12
  • $\begingroup$ Related: math.stackexchange.com/questions/317706 $\endgroup$ – saz Feb 25 '14 at 19:53