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Supposing $Y_1,Y_2,\cdots, Y_n$ be random variables such that $Y_i \in \mathcal{L}^2(\Omega,\Sigma,P)$ for all $i$. What are the conditions under which $$\mathrm{span}(Y_1,Y_2,\cdots,Y_n) = \mathcal{L}^2(\Omega,\sigma(Y_1,Y_2,\cdots,Y_n),P) ?$$

I think that this should be true when $Y_1,Y_2,\cdots,Y_n$ are jointly Gaussian, but even in this case, a proof would be highly appreciated.

Thanks, Phanindra

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  • $\begingroup$ What do you mean by $\mathrm{span}(Y_1,Y_2,\cdots,Y_n)$? $\endgroup$ – Ashok Mar 24 '12 at 12:35
  • $\begingroup$ Random variables of the form $\sum_{i=1}^n a_i Y_i$, where $a_i \in \mathbb{R}$ $\endgroup$ – jpv Mar 24 '12 at 12:44
  • $\begingroup$ I don't think this can be true (even in the Gaussian case). Because then $\mathcal{L}^2$ would become a finite dimensional space, but which is not the case, no? $\endgroup$ – Ashok Mar 27 '12 at 8:06
  • $\begingroup$ @Ashok: You were right. I just got an answer to the same question on mathoverflow. I was not aware that $L^2$ is infinite dimensional. $\endgroup$ – jpv Mar 30 '12 at 9:59

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