$\varepsilon_{ij} \in \mathcal{N}(0,\sigma^2)$. What can we say about the third moment?

Question

the errors $\varepsilon_{ij}$ are independent $\mathcal{N}(0,\sigma^2)$ random variables. What can we say about the third moment? For example, is $\mathbb{E}(\varepsilon_{ij}^3)$ bounded?

I really appreciate any help you can provide.

Answer 1

Hint:

The function $$x\mapsto \frac{1}{\sqrt{2\pi}\sigma }x^{2n+1}e^{-\frac{x^2}{2\sigma ^2}},$$ is odd for all $n$.