Timeline for If $X∼F_1$, $Y∼F_2$, under what conditions on $F_1$, $F_2$ can we construct $Y=E(X\mid\mathscr{G})$ for some $\mathscr{G}$?

8 events

when toggle format	what		by	license	comment
Oct 19, 2015 at 5:17	vote	accept	Anthony Lee Zhang
Oct 18, 2015 at 18:34	answer	added	zhoraster		timeline score: 7
Oct 18, 2015 at 6:00	comment	added	Christian Remling		I of course forgot to divide by $\sum_k w_{jk}$ above.
Oct 18, 2015 at 5:38	comment	added	Christian Remling		There is an obvious, semi-trivial answer to this for discrete distributions: you can split each probability $P(X=x_k)$ at will, say $P(X=x_k)=\sum_j w_{jk}$, and then do your partial averages to obtain $Y$, which will take the values $y_j=\sum_k w_{jk}x_k$ with the obvious probabilities. Essentially, this should tell the whole story. You can do similar things with disintegrations of measures in general (though there are measurability issues, as always).
Oct 16, 2015 at 2:14	comment	added	Nate Eldredge		Obvious necessary conditions: $\int x\,F_1(dx) = \int x\,F_2(dx)$, and $\int \varphi(x)\,F_1(dx) \le \int \varphi(x)\,F_2(dx)$ for all $\varphi$ convex and bounded below (by conditional Jensen).
Oct 16, 2015 at 1:45	history	edited	Bjørn Kjos-Hanssen	CC BY-SA 3.0	edited title
Oct 16, 2015 at 0:21	review	First posts
Oct 16, 2015 at 1:13
Oct 16, 2015 at 0:18	history	asked	Anthony Lee Zhang	CC BY-SA 3.0