Timeline for Why does the dispersion of X about its conditional mean decreases as the σ−algebra grows? [closed]

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Nov 25, 2020 at 18:30	history	closed	user44191 coudy Desiderius Severus LeechLattice ARG		Not suitable for this site
Nov 10, 2020 at 5:27	vote	accept	Haosheng Zhou
Nov 10, 2020 at 5:16	comment	added	Haosheng Zhou		Thanks for your help!
Nov 9, 2020 at 15:02	comment	added	Nate Eldredge		A useful fact is that for square-integrable $X$, $\mathbb{E}[X \mid \mathcal{G}]$ is the $L^2$ orthogonal projection of $X$ onto the subspace of $\mathcal{G}$-measurable random variables. In particular, $\mathbb{E}[X \mid \mathcal{G}]$ minimizes the $L^2$ distance to that subspace, so if $Y$ is any other $\mathcal{G}$-measurable random variable, then $\mathbb{E}[(X - \mathbb{E}[X \mid \mathcal{G}])^2] \le E[(X-Y)^2]$. Now applying this with $\mathcal{G} = \mathcal{G}_2$ and $Y = \mathbb{E}[X \mid \mathcal{G}_1]$ you should get the result.
Nov 9, 2020 at 14:51	answer	added	Iosif Pinelis		timeline score: 0
Nov 9, 2020 at 13:29	review	Suggested edits
Nov 9, 2020 at 14:16
Nov 9, 2020 at 12:08	review	Close votes
Nov 25, 2020 at 18:30
Nov 9, 2020 at 10:47	review	First posts
Nov 9, 2020 at 11:51
Nov 9, 2020 at 10:40	history	asked	Haosheng Zhou	CC BY-SA 4.0