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I want to show a sequence of normal $N(\mu_n,\sigma_n^2)$ distributed random variables is tight iff sequences $\mu_n,\sigma_n^2$ are bounded.

Here a sequence $(\nu_n)$ of probability measures is tight if $\lim_{M\to\infty}\inf_n \nu_n[-M,M] = 1$, which in this case means

$$\lim_{M\to\infty}\inf_n\frac{1}{\sigma_n\sqrt{2\pi}}\int_{-M}^M \exp\left(-\frac{1}{2}\left(\frac{x-\mu_n}{\sigma_n}\right)^2\right)d(x)=1.$$

I find this hard to see...

Is there a way we can deduce this by looking at the characterstic functions $$\Phi_n(u)=\exp(\mu_n u i -\sigma_n^2 u^2/2)$$

Is there maybe an equivalent tightness condition for characteristic functions..? Thanks for any help.

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Possible duplicate – Stefan Hansen Nov 26 '12 at 14:34

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