Let $X_n$ a positive submartingale. What I don't understand is why $$E[|X_n|]$$ is increasing in n.
Could someone help me to understand it?
2026-03-30 08:04:39.1774857879
E[$X_n$] of submartingale is increasing
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By the definition of submartingale, $E(X_{n+1} \vert \mathcal{F}_n) \ge X_n$. You take expectation on both sides of the inequality. Then you get $$E(X_{n+1}) = E\left(E(X_{n+1} \vert \mathcal{F}_n)\right) \ge E(X_n).$$