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I am looking for an example that shows we can't drop the continuity assumption on the limiting characteristic function in Levy's continuity theorem.

I got some hints from my professor that use a sequence of normal distributions {$ \mu_n$ } with $ \mu_n$ having mean 0 and variance n.

Could anyone help me out with how this construction works?

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    $\begingroup$ So what is the limit of the characteristic functions of the $\mu_n$ in the professor's hint. Is it continuous? Is it the sequence $\mu_n$ tight? $\endgroup$ – kimchi lover Aug 3 '17 at 22:51
  • $\begingroup$ Why can't you follow the precise suggestion by your teacher? (Unrelated: Are you still using two accounts on mse?) $\endgroup$ – Did Aug 6 '17 at 21:36

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