A Submartingale has uniformly Bounded First Moment iff it has uniformly Bounded Positive Moment

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space

($\Rightarrow$): Because $(X_{n}){n \geq 1}$ is a Submartingale

Thus, $$\begin{align*} \sup\limits_{n \geq 1} \mathbb{E}|X_{n}| &= \sup\limits_{n \geq 1} (\mathbb{E}(X_{n})^{+} + \mathbb{E}(X_{n})^{-})\\ &\leq \sup\limits_{n \geq 1} (\mathbb{E}(X_{n})^{+}) + \sup\limits_{n \geq 1} (\mathbb{E}(X_{n})^{-})\\ &< \infty \end{align*}$$ $\checkmark$

($\Leftarrow$): This follows because $$\infty > \sup\limits_{n \geq 1} \mathbb{E}|X_{n}| \geq \sup\limits_{n \geq 1} (\mathbb{E}(X_{n})^{+} + \mathbb{E}(X_{n})^{-}) \geq \sup\limits_{n \geq 1} \mathbb{E}(X_{n})^{+}$$ $\checkmark$ $\blacksquare$