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.


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 $g\colon x\mapsto \max\left\{x,0\right\}$.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.