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world:hewitt-savage
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world:hewitt-savage [2026/02/05 15:20] rdouc |
world:hewitt-savage [2026/02/06 13:06] (current) rdouc |
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| **Key Idea:** | **Key Idea:** | ||
| - | We aim to establish the identity \(\mathbb{P}(A) = \mathbb{E}[\mathsf{1}_A \mathsf{1}_A] = \mathbb{E}[\mathsf{1}_A]^2 = \mathbb{P}(A)^2\). | + | We aim to establish the identity \(\mathbb{P}(A) = \mathbb{E}[\mathsf{1}_A \mathsf{1}_A] = \mathbb{E}[\mathsf{1}_A]^2 = \mathbb{P}(A)^2\). To do so, the idea is to approximate $\mathsf{1}_A$ by the indicators of two different independent events. |
| - | + | ||
| **Step 1: Approximation Lemma** | **Step 1: Approximation Lemma** | ||
| Line 61: | Line 59: | ||
| **Step 3: Independence and Expectation** | **Step 3: Independence and Expectation** | ||
| Introduce the intermediate quantities \(\mathsf{1}_{\bar{B}}(X_{1:n})\) and \(\mathsf{1}_{\bar{B}}(X_{n+1:2n})\) which are independent by independence of \((X_i)\). Then, we obtain: | Introduce the intermediate quantities \(\mathsf{1}_{\bar{B}}(X_{1:n})\) and \(\mathsf{1}_{\bar{B}}(X_{n+1:2n})\) which are independent by independence of \((X_i)\). Then, we obtain: | ||
| - | \[ | + | $$ |
| - | |\mathbb{E}[\mathsf{1}_A \mathsf{1}_A] - \mathbb{E}[\mathsf{1}_A]\mathbb{E}[\mathsf{1}_A]| \leq 4\delta. | + | |\mathbb{E}[\mathsf{1}_A \mathsf{1}_A] - \mathbb{E}[\mathsf{1}_A]\mathbb{E}[\mathsf{1}_A]| \leq |\mathbb{E}[\mathsf{1}_A \mathsf{1}_A] - \mathbb{E}[\mathsf{1}_{\bar{B}}(X_{1:n})\mathsf{1}_{\bar{B}}(X_{n+1:2n})]| + | \mathbb{E}[\mathsf{1}_{\bar{B}}(X_{1:n})]\mathbb{E}[\mathsf{1}_{\bar{B}}(X_{n+1:2n})]- \mathbb{E}[\mathsf{1}_A] \mathbb{E} [\mathsf{1}_A]| \leq 4\delta. |
| - | \] | + | $$ |
| Since \(\delta > 0\) is arbitrary, we conclude that: | Since \(\delta > 0\) is arbitrary, we conclude that: | ||
| \[ | \[ | ||
world/hewitt-savage.1770301241.txt.gz · Last modified: 2026/02/05 15:20 by rdouc
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