4.4. Martingale inequalities and convergence in $L^p$

probability Durrett

In this section we look into the condition that makes a martingale converges in $L^p$, $p>1$ in detail. We start by proving Doob’s inequality. By using this result we prove martingale inequalities which will then be used to prove Doob’s $L^p$ maximal inequality. $L^p$ convergence is direct from them. Lastly, as an extension of Doob’s inequality, I will briefly state a version of optional stopping.
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