202502190102

Tags : Intro to Martingales

Doob’s Upcrossings Inequality

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Theorem

Let $X$ be a sub-martingale. Then for $\alpha <\beta$

$$E[U_m(\{X_j,1\le j\le m\},\alpha ,\beta )]\le \frac{E[|X_m|+|\alpha |]}{\beta -\alpha }$$

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References