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If $X_n$ is a sequence of random variables converging in distribution to a chi-squared distribution with degree of freedom being $\nu$, will $\sqrt{X_n}$ converges in distribution to the sqrt of the squared sum of $\nu$ iid standard normal random variables? Thanks!

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$\{Xn\}$ converges in distribution to $X $if and only if $\mathbf{E}f(Xn) → \mathbf{E}ƒ(X)$ for all bounded, continuous functions ƒ;


So it is true.

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