If $$X_1 = 1,\quad X_{n+1} = 2X_n \text{ or }0 \text{ with equal probability }(P=1/2),$$ with mean: $E|Xn|=1$
How can I prove this is a martingale?
2026-03-31 22:25:49.1774995949
How can I prove this is a martingale?
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Denotes $M_{n+1}$ the set on which the random variable $X_{n+1}$ is zero. With the natural filtration $\mathcal{F}_n$ you get $$E(X_{n+1}|\mathcal{F}_n)=E(1_{M_{n+1}} \cdot 0|\mathcal{F}_n)+E(1_{M_{n+1}^\complement}\cdot 2X_n|\mathcal{F}_n) = 2X_n\cdot E(1_{M_{n+1}^\complement})=X_n.$$