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I'm stuck with a problem of probability. Suppose $\{N_n,n \geq 0\}$ is a sequence of normal random variables. Show $N_n \Rightarrow N_0$ iff $$E(N_n) \to E(N_0)$$ and $$Var(N_n) \to Var(N_0).$$ I'm done when I suppose the convergence of the expected value and the variance, but i'm not able to make a proof when I suppose $N_n \Rightarrow N_0$.

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  • $\begingroup$ Can you clarify your comment? Since the normal distribution is entirely characterized by its mean and variance, and convergence in distribution is a pretty weak convergence, it almost seems self-evident that this is true. What did you mean by "I'm not able to make a proof..." $\endgroup$ – eSurfsnake May 28 '19 at 0:34
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Weak convergence is equivalent to convergence of characteristic functions. Take absolute value in characteristic functions to see that the variances converge. Then it becomes obvious that the means also converge.

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  • $\begingroup$ I see... this is the Continuity theorem? $\endgroup$ – Bayesian guy May 28 '19 at 1:45
  • $\begingroup$ $e^{itx}$ is a bounded continuous function so the characteristic functions converge. $\endgroup$ – Kavi Rama Murthy May 28 '19 at 11:33

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