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world:de-finetti [2026/02/05 15:08] rdouc |
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| - | on{{page>:defs}} | + | {{page>:defs}} |
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| <WRAP center round tip 80%> | <WRAP center round tip 80%> | ||
| **De Finetti's Theorem:** | **De Finetti's Theorem:** | ||
| - | Let $(X_i)_{i\in\mathbb{N}}$ be a family of **exchangeable random elements** defined on a measurable space $(\mathsf{X},\mathcal{X})$. Then, there exists a $\sigma$-field $\mathcal{G}_\infty$ such that, **conditionally on $\mathcal{G}_\infty$**, the random variables $(X_i)_{i\in\mathbb{N}}$ are **independent and identically distributed (i.i.d.)**. | + | Let $(X_i)_{i\in\mathbb{N}}$ be a family of **exchangeable random elements** taking values on a measurable space $(\mathsf{X},\mathcal{X})$. Then, there exists a $\sigma$-field $\mathcal{G}_\infty$ such that, **conditionally on $\mathcal{G}_\infty$**, the random variables $(X_i)_{i\in\mathbb{N}}$ are **independent and identically distributed (i.i.d.)**. |
| </WRAP> | </WRAP> | ||
| Line 49: | Line 49: | ||
| \mathbb{E}[f(X_1,\ldots,X_k)\mid\mathcal{G}_\infty] | \mathbb{E}[f(X_1,\ldots,X_k)\mid\mathcal{G}_\infty] | ||
| =\lim_{n\to\infty}\frac1{n(n-1)\cdots(n-k+1)} | =\lim_{n\to\infty}\frac1{n(n-1)\cdots(n-k+1)} | ||
| - | \sum_{\substack{i_{1:k} \in [1:n]^k \le n,\ldots,1 \le i_k\le n\\ i_j\neq i_\ell}} | + | \sum_{\substack{i_{1:k} \in [1:n]^k\\ i_j\neq i_\ell}} |
| f(X_{i_1},\ldots,X_{i_k}). | f(X_{i_1},\ldots,X_{i_k}). | ||
| $$ | $$ | ||
| Line 55: | Line 55: | ||
| $$ | $$ | ||
| \frac1{n(n-1)\cdots(n-k+1)} | \frac1{n(n-1)\cdots(n-k+1)} | ||
| - | \sum_{\substack{1\le i_1,\ldots,i_k\le n\\ i_j\neq i_\ell}} | + | \sum_{\substack{i_{1:k} \in [1:n]^k\\ i_j\neq i_\ell}} |
| f(X_{i_1},\ldots,X_{i_k}) + O\lr{\frac1n} = \frac1{n^k}\sum_{i_1=1}^n\cdots\sum_{i_k=1}^n f(X_{i_1},\ldots,X_{i_k}). | f(X_{i_1},\ldots,X_{i_k}) + O\lr{\frac1n} = \frac1{n^k}\sum_{i_1=1}^n\cdots\sum_{i_k=1}^n f(X_{i_1},\ldots,X_{i_k}). | ||
| $$ | $$ | ||
| Line 63: | Line 63: | ||
| =\prod_{\ell=1}^k\mathbb{E}[f_\ell(X_1)\mid\mathcal{G}_\infty]. | =\prod_{\ell=1}^k\mathbb{E}[f_\ell(X_1)\mid\mathcal{G}_\infty]. | ||
| $$ | $$ | ||
| - | * Thus, conditionally on $\mathcal{G}_\infty$, $(X_i)$ are independent. | + | * Thus, conditionally on $\mathcal{G}_\infty$, $(X_i)$ are independent. $\blacksquare$ |
| - | ==== Comments: convergence of Empirical Averages for i.i.d. Random Variables ==== | + | <WRAP center round todo 80%> |
| + | If $(X_i)$ are real-valued random variables, then, since the distribution of any random variable $X$ is completely determined by the values $\mathbb{P}(X \le x)$ for $x \in \mathbb{Q}$, we can replace $\mathcal{G}_\infty$ with the $\sigma$-field generated by the countable family of random variables | ||
| + | \[ | ||
| + | \left\{ \mathbb{E}\big[\mathbf{1}_{\{X_1 \le x\}} \mid \mathcal{G}_\infty \big] : x \in \mathbb{Q} \right\}. | ||
| + | \] | ||
| + | It follows that there exists a random variable $S$ such that, conditional on $S$, the sequence $(X_i)$ is independent and identically distributed. | ||
| + | |||
| + | </WRAP> | ||
| + | |||
| + | |||
| + | ==== Comments: Another proof of the Law of Large Numbers for i.i.d. Random Variables ==== | ||
| - | The previous approach allows to prove the LLN. | + | The previous approach allows to prove the strong law of large numbers (for a proof of the LLN using only the dominated convergence theorem, [[world:lln| click here]]) |
| <WRAP center round tip 80%> | <WRAP center round tip 80%> | ||
| Line 77: | Line 87: | ||
| - | **Reverse Martingale Convergence Theorem:** | + | **Proof:** |
| - | By the **reverse martingale convergence theorem**, the empirical average converges almost surely to the conditional expectation with respect to the tail σ-field \(\mathcal{G}_\infty\): | + | By the **reverse martingale convergence theorem** (see [[world:forward-downward-martingale|Upcrossing Inequality and Martingale Convergence]]), the empirical average converges almost surely to the conditional expectation with respect to the tail σ-field \(\mathcal{G}_\infty\): |
| \[ | \[ | ||
| \frac{1}{n} \sum_{i=1}^n X_i = \mathbb{E}[X_1|\mathcal{G}_n] \xrightarrow{a.s.} \mathbb{E}[X_1|\mathcal{G}_\infty]. | \frac{1}{n} \sum_{i=1}^n X_i = \mathbb{E}[X_1|\mathcal{G}_n] \xrightarrow{a.s.} \mathbb{E}[X_1|\mathcal{G}_\infty]. | ||
| \] | \] | ||
| - | **Hewitt-Savage 0-1 Law:** | ||
| The [[world:hewitt-savage|Hewitt-Savage 0-1 Law]] states that any \(\mathcal{G}_\infty\)-measurable random variable is almost surely constant. Consequently, the conditional expectation \(\mathbb{E}[X_1|\mathcal{G}_\infty]\) is almost surely equal to its unconditional expectation: | The [[world:hewitt-savage|Hewitt-Savage 0-1 Law]] states that any \(\mathcal{G}_\infty\)-measurable random variable is almost surely constant. Consequently, the conditional expectation \(\mathbb{E}[X_1|\mathcal{G}_\infty]\) is almost surely equal to its unconditional expectation: | ||
| \[ | \[ | ||
| Line 89: | Line 98: | ||
| \] | \] | ||
| - | **Conclusion:** | ||
| Thus, the empirical average converges almost surely to the theoretical expectation: | Thus, the empirical average converges almost surely to the theoretical expectation: | ||
| \[ | \[ | ||
world/de-finetti.1770300501.txt.gz · Last modified: 2026/02/05 15:08 by rdouc
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