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 . Let $(X_{n}){n \geq 1}$ be a discrete-time Submartingale wrt Discrete-Time Filtration $\mathcal{F}{*} := (\mathcal{F}{n} \subset \mathcal{B}){n \in \mathbb{N}}$ . Then $\sup\limits_{n \geq 1} \mathbb{E}(X_{n})^{+} < \infty$ If and Only If $\sup\limits_{n \geq 1} \mathbb{E}|X_{n}| < \infty$.

($\Rightarrow$): Because $(X_{n}){n \geq 1}$ is a Submartingale and Submartingales have Non-Decreasing Expectation, $\forall n \geq 1$, $$\begin{align*} &\mathbb{E}(X{n}) \geq \mathbb{E}(X_{1})\\ \Rightarrow&\mathbb{E}(X_{n})^{+} - \mathbb{E}(X_{n})^{-} \geq \mathbb{E}(X_{1})\\ \Rightarrow&\mathbb{E}(X_{n})^{+} - \mathbb{E}(X_{1}) \geq \mathbb{E}(X_{n})^{-} \\ \Rightarrow&\infty > \sup\limits_{n \geq 1} \mathbb{E}(X_{n})^{+} - \mathbb{E}(X_{1}) \geq \sup\limits_{n \geq 1} \mathbb{E}(X_{n})^{-} \end{align*}$$

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