Generalisation of strong law of large numbers for kth moments

Question

I have a quick question regarding the SLLN, and haven't been lucky enough to find the answer to this after searching on the internet but will ask on this forum:

I am aware that the SLLN states that for a sequence of R.V.s the sample average converges a.s. to its expected value as $n \rightarrow \infty$i.e. $\frac{1}{n}\sum_{i=1}^{n}X_i \rightarrow \mathbb{E}[X]$

I just wanted to know if this generalises to the kth raw moment? i.e. $\frac{1}{n}\sum_{i=1}^{n}X_i^k \rightarrow \mathbb{E}[X^k]$

assuming that the k-th moment actually exists and is finite?

Any responses to the above would be very much appreciated.

Answer 1

Let $Y = X^k$

then apply apply the law of large numbers to $Y$