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Suppose that the sequence of random variables $\{X_{n}\}$ converges to a random variable $X$ in distribution, and a numeric sequence $a_n$ converges to $1$. Prove that $a_nX_n$ converges to $X$ in distribution.

I'm not quite sure how to approach this problem. I recently learned about Weak Convergence, but that seems to be no help here. For example, I am aware of Prokhorov's Theorem, but that seems to be useless here. So I am instead trying to use different inequalities, like Markov's, Chebyshev, etc. I'm studying for my exam and will greatly appreciate your help in solving the problem.

Thank you

  • What have you tried? Such a statement seems almost obvious, E.g. if we replace "random variables" with "complex numbers", the convergence is immediate. So, have you tried just writing it out? – Calvin Lin Dec 01 '19 at 18:30
  • I as going to answer this question, but if it is a duplicate then I will defer to your wisdom, @saz. – Math1000 Dec 01 '19 at 18:34

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