Martingale to power

Question

Assume a Martingale $(X_i)$. My Question is what do I know about $(X_i^p)$ with $p\in \Bbb{N}_{\geq 2}$? I understand that I need $X_i\in\mathcal{L}^p$ to have any sort of (Sub-/Super-)Martingale.

Answer 1

Assume that $E|X_i|^{p} <\infty$ for all $n$. If $(X_i)$ is a martingale then $(|X_i|)$ is a sub-martingale. By Jensen's inequality for conditional expectations we see that $(X_i^{p})$ is a sub-martingale if $p$ is even. For $p$ odd, $(X_i^{p})$ need not be a sub-martingale or a super-martingale.