# Martingale to power

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.

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.