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world:definetti [2026/02/03 13:44] rdouc [Proof] |
world:definetti [2026/02/03 19:01] (current) rdouc |
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| + | on{{page>:defs}} | ||
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| ====== De Finetti's Representation Theorem for Exchangeable Random Elements ====== | ====== De Finetti's Representation Theorem for Exchangeable Random Elements ====== | ||
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| </WRAP> | </WRAP> | ||
| + | The proof is based on the paper "Uses of exchangeability" by J. F. Kingman [[https://projecteuclid.org/journalArticle/Download?urlId=10.1214%2Faop%2F1176995566|Click here to see the paper]]. | ||
| ===== Proof ===== | ===== Proof ===== | ||
| Without loss of generality, we model $(X_i)_{i\in\mathbb{N}}$ as the coordinate projections on the canonical probability space $(\mathsf{X}^{\mathbb{N}},\mathcal{X}^{\otimes\mathbb{N}},\mathbb{P})$. We proceed as follows: | Without loss of generality, we model $(X_i)_{i\in\mathbb{N}}$ as the coordinate projections on the canonical probability space $(\mathsf{X}^{\mathbb{N}},\mathcal{X}^{\otimes\mathbb{N}},\mathbb{P})$. We proceed as follows: | ||
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| $$ | $$ | ||
| \mathbb{E}\!\left[\left(\frac1n\sum_{i=1}^n h(X_i)\right)\mathbf1_A\right] | \mathbb{E}\!\left[\left(\frac1n\sum_{i=1}^n h(X_i)\right)\mathbf1_A\right] | ||
| + | =\frac1n\sum_{i=1}^n \mathbb{E}\!\left[\left( h(X_i)\right)\mathbf1_A\right] | ||
| =\mathbb{E}[h(X_1)\mathbf1_A]. | =\mathbb{E}[h(X_1)\mathbf1_A]. | ||
| $$ | $$ | ||
| - | * Hence | + | * The two previous item show the amazing formula: |
| $$ | $$ | ||
| \frac1n\sum_{i=1}^n h(X_i)=\mathbb{E}[h(X_1)\mid\mathcal{G}_n], \quad a.s. | \frac1n\sum_{i=1}^n h(X_i)=\mathbb{E}[h(X_1)\mid\mathcal{G}_n], \quad a.s. | ||
| $$ | $$ | ||
| - | * By the reverse martingale convergence theorem, | + | * By the **reverse martingale convergence** theorem (see for example [[world:forward-downward-martingale|Upcrossing Inequality and Martingale Convergence]]), |
| $$ | $$ | ||
| \frac1n\sum_{i=1}^n h(X_i)\xrightarrow{\mathrm{a.s.}}\mathbb{E}[h(X_1)\mid\mathcal{G}_\infty]. | \frac1n\sum_{i=1}^n h(X_i)\xrightarrow{\mathrm{a.s.}}\mathbb{E}[h(X_1)\mid\mathcal{G}_\infty]. | ||
| Line 44: | 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{1\le i_1,\ldots,i_k\le n\\ i_j\neq i_\ell}} | + | \sum_{\substack{i_{1:k} \in [1:n]^k \le n,\ldots,1 \le i_k\le n\\ i_j\neq i_\ell}} |
| f(X_{i_1},\ldots,X_{i_k}). | f(X_{i_1},\ldots,X_{i_k}). | ||
| $$ | $$ | ||
| Line 51: | Line 56: | ||
| \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{1\le i_1,\ldots,i_k\le n\\ i_j\neq i_\ell}} | ||
| - | f(X_{i_1},\ldots,X_{i_k}) + 0\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}). |
| $$ | $$ | ||
| * Hence, for product functions $f(x_1,\ldots,x_k)=f_1(x_1)\cdots f_k(x_k)$, | * Hence, for product functions $f(x_1,\ldots,x_k)=f_1(x_1)\cdots f_k(x_k)$, | ||
world/definetti.1770122647.txt.gz · Last modified: 2026/02/03 13:44 by rdouc
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