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Suppose $Z_n$ is a sequence of random variables that converge to standard normal in distribution. I saw the claim that $Z_n^2$ converge to chi-square random variable in distribution. But why is that true? How do you prove this?
The only way that I know we can end up with something converges to chi square in distribution is second order delta method, but that doesn't seems to be relevant here.
This follows that the fact that if $Z_n\to Z$ in distribution, and $h$ is a continuous function, then $h(Z_n)\to h(Z)$ in distribution (a hint for the proof of this fact is here https://math.stackexchange.com/questions/1915776/continuous-function-keeps-the-convergence-in-distribution).
