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world:useful-bounds [2022/11/17 17:35] rdouc [Doob's inequalities] |
world:useful-bounds [2022/11/17 17:41] rdouc [Doob's inequalities] |
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\epsilon \PP\lr{\max_{k=1}^n X_k \geq \epsilon} \leq \PE[X_{\tau_\epsilon} \indiacc{\tau_\epsilon \leq n}] \leq \PE[X_{\tau_\epsilon \wedge n}]\leq \PE[X_{\tau_\epsilon \wedge 0}]=\PE[X_0] | \epsilon \PP\lr{\max_{k=1}^n X_k \geq \epsilon} \leq \PE[X_{\tau_\epsilon} \indiacc{\tau_\epsilon \leq n}] \leq \PE[X_{\tau_\epsilon \wedge n}]\leq \PE[X_{\tau_\epsilon \wedge 0}]=\PE[X_0] | ||
$$ | $$ | ||
- | where we used that $(X_{\tau_\epsilon \wedge n})$ is an integrable super-martingale. | + | where we used in the second inequality that $(X_n)$ is non-negative and in the third inequality that $(X_{\tau_\epsilon \wedge n})$ is a supermartingale. |
* We now turn to the proof of **(ii)**. Using \eqref{eq:fond}, | * We now turn to the proof of **(ii)**. Using \eqref{eq:fond}, | ||
\begin{align*} | \begin{align*} |