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| ==== Hewitt-Savage 0-1 Law ==== | ==== Hewitt-Savage 0-1 Law ==== | ||
| - | Let \((X_i)_{i\in\mathbb{N}}\) be a family of independent and identically distributed (i.i.d.) random variables defined on a common probability space and taking values in a measurable space \((\mathsf{X}, \mathcal{X})\). | + | Let \((X_i)_{i\in\mathbb{N}}\) be a family of **independent and identically distributed (i.i.d.)** random variables defined on a common probability space and taking values in a measurable space \((\mathsf{X}, \mathcal{X})\). |
| **Equivalent Formulation:** | **Equivalent Formulation:** | ||
| - | We can also consider \(X_i\) as the coordinate projection associated with the probability space \((\mathsf{X}^\mathbb{N}, \mathcal{X}^{\otimes\mathbb{N}}, \mathbb{P})\), where, under \(\mathbb{P}\), the sequence \((X_i)\) is i.i.d. | + | We can also consider \(X_i\) as the coordinate projection associated with the probability space \((\mathsf{X}^\mathbb{N}, \mathcal{X}^{\otimes\mathbb{N}}, \mathbb{P})\), where, under \(\mathbb{P}\), the sequence \((X_i)\) is **i.i.d**. |
| **Permutation-Invariant σ-Fields:** | **Permutation-Invariant σ-Fields:** | ||
| - | For any \(n \in \mathbb{N}\), let \(\mathcal{G}_n\) be the σ-field generated by measurable functions \(f: \mathsf{X}^\mathbb{N} \to \mathbb{R}\) that are invariant under any permutation of the first \(n\) coordinates. Define the tail σ-field as: | + | For any \(n \in \mathbb{N}\), let \(\mathcal{G}_n\) be the σ-field generated by measurable functions \(f: \mathsf{X}^\mathbb{N} \to \mathbb{R}\) that are invariant under any permutation of the first \(n\) coordinates. Define the σ-field $\mathcal{G}_\infty$ as: |
| \[ | \[ | ||
| \mathcal{G}_\infty = \bigcap_{n\in\mathbb{N}} \mathcal{G}_n, | \mathcal{G}_\infty = \bigcap_{n\in\mathbb{N}} \mathcal{G}_n, | ||
| Line 13: | Line 13: | ||
| which is generated by measurable functions invariant under any permutation of a finite number of coordinates. | which is generated by measurable functions invariant under any permutation of a finite number of coordinates. | ||
| - | **Hewitt-Savage 0-1 Law Statement:** | + | <WRAP center round tip 80%> |
| - | We will prove that for any \(A \in \mathcal{G}_\infty\), \(\mathbb{P}(A) = 0\) or \(1\). This result is known as the **Hewitt-Savage 0-1 Law**. | + | **Statement:** |
| + | For any \(A \in \mathcal{G}_\infty\), \(\mathbb{P}(A) = 0\) or \(1\). | ||
| + | |||
| + | This result is known as the **Hewitt-Savage 0-1 Law**. | ||
| + | |||
| + | </WRAP> | ||
| - | --- | ||
| ==== Proof of the Hewitt-Savage 0-1 Law ==== | ==== Proof of the Hewitt-Savage 0-1 Law ==== | ||
| Line 24: | Line 28: | ||
| **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** | ||
| - | Let \(\delta > 0\). By the approximation lemma, there exist \(n \in \mathbb{N}\) and a set \(B \in \mathcal{F}_n = \sigma(X_{1:n})\) such that: | + | Let \(\delta > 0\). By the [[world:approximation-lemma|Approximation Lemma]], there exist \(n \in \mathbb{N}\) and a set \(B \in \mathcal{F}_n = \sigma(X_{1:n})\) such that: |
| \[ | \[ | ||
| \mathbb{E}\left[|\mathsf{1}_A - \mathsf{1}_B|\right] \leq \delta. | \mathbb{E}\left[|\mathsf{1}_A - \mathsf{1}_B|\right] \leq \delta. | ||
| Line 42: | Line 44: | ||
| \] | \] | ||
| - | --- | ||
| **Step 2: Permutation Argument** | **Step 2: Permutation Argument** | ||
| Line 54: | Line 55: | ||
| \] | \] | ||
| - | --- | + | |
| **Step 3: Independence and Expectation** | **Step 3: Independence and Expectation** | ||
| - | Introduce the intermediate quantities \(\mathsf{1}_{\bar{B}}(X_{1:n})\) and \(\mathsf{1}_{\bar{B}}(X_{n+1:2n})\). By the independence of \((X_i)\) and the invariance of \(A\), 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: | ||
| \[ | \[ | ||
| Line 67: | Line 68: | ||
| which implies \(\mathbb{P}(A) = \mathbb{P}(A)^2\). Therefore, \(\mathbb{P}(A) \in \{0,1\}\). | which implies \(\mathbb{P}(A) = \mathbb{P}(A)^2\). Therefore, \(\mathbb{P}(A) \in \{0,1\}\). | ||
| - | --- | + | |
| **Conclusion:** | **Conclusion:** | ||
| The Hewitt-Savage 0-1 Law states that any permutation-invariant event in the tail σ-field \(\mathcal{G}_\infty\) has probability 0 or 1. | The Hewitt-Savage 0-1 Law states that any permutation-invariant event in the tail σ-field \(\mathcal{G}_\infty\) has probability 0 or 1. | ||
world/hewitt-savage.1770291367.txt.gz · Last modified: 2026/02/05 12:36 by rdouc
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