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Suppose $X_1,\ldots,X_n$ are iid random variables with mean $0$ and variance $1$ . By the CLT we know that $\frac{1}{\sqrt{n}}\sum_{i=1}^n X_i$ converges in distribution to a standard normal distribution.

  1. Can you infer from the CLT that $\left(\frac{1}{\sqrt{n}}\sum_{i=1}^n X_i \right)^2$ converges in distribution to Chi-Squared with 1 degree of freedom?
  2. Can you infer that $\max\left\{ 0,\frac{1}{\sqrt{n}}\sum_{i=1}^n X_i\right\}$ converges in distribution to the distribution of $\max\left\{ 0,Z\right\}$ where $Z\sim\text{Normal}\left(0,1\right)$ ?

I'm led to believe both of these things are true but I can't manage to justify to myself why. Help would be appreciated.

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As pointed out by d.k.o., everything holds as a consequence of the continuous mapping theorem, which states the following: if $\left(Z_n\right)_{n\geqslant 1}$ is a sequence of (real-valued) random variables which converges in distribution to $Z$, then for all continuous function $g\colon\mathbb R\to\mathbb R$, the sequence $\left(g\left(Z_n\right)\right)_{n\geqslant 1}$ converges in distribution to $g\left(Z\right)$.

  1. Apply this to $g\colon x\mapsto x^2$.
  2. Apply this to $g\colon x\mapsto \max\left\{x,0\right\}$.
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