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world:definetti [2026/02/03 14:06] rdouc [Proof] |
world:definetti [2026/02/03 19:01] (current) rdouc |
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| - | {{page>:defs}} | + | on{{page>:defs}} |
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| \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 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{1\le i_1 \le n,\ldots,1 \le 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 56: | 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.1770123975.txt.gz · Last modified: 2026/02/03 14:06 by rdouc
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