Convergence in probability of $\frac1n\sum_{i=1}^n X_i^2$ where $X_i$'s are i.i.d $N(0,1)$

Question

Let $X_1,X_2,...$ be a sequence of i.i.d $N(0,1)$ RVs. Then as $ n \rightarrow \infty$ , $\frac { 1 } { n } \sum\limits _ { i = 1 } ^ { n } X _ { i } ^ { 2 }$ converges in probability to ...?

I know sum of squares of N(0,1) will be chi square with n d.f but I am not able to figure out the value of probability.

Any help would be much appreciated.

Answer 1

$\{X_i^{2}\}$ is also i.i.d. Just apply Law of Large Numbers.

The limit is $\mathbb{E}\left[X_1^{2} \right]=1$.