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world:hewitt-savage
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world:hewitt-savage [2026/02/06 13:06] rdouc [Proof of the Hewitt-Savage 0-1 Law] |
world:hewitt-savage [2026/02/06 13:06] (current) rdouc |
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| **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 |\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. | |\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.1770379577.txt.gz · Last modified: 2026/02/06 13:06 by rdouc
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