I am working with conditional expectations and am trying to derive a limit property.
Consider $(_)_{\in\mathbb N}$ a sequence of square integrable random variables, that converge in $L^2$ to a square integrable random variable $Y$. Additionally assume that $\mathbb E[Y_n\mid Y]=Y_n$ (for example, $Y_n$ is a sequence of discrete quantizers of $Y$).
Is there anyway at all of guaranteeing that for some other $X$ in $L^2$, and for some form of convergence ($L^2, \mathbb P$ etc.) : $$\lim_{n\to+\infty}\mathbb E[X\mid Y_n]=\mathbb E[X\mid Y].$$
I am aware of the following similar question :
Conditional expectation of asymptotically independent random variables
but in that case $\mathbb E[Y_n\mid Y]=Y_n$ does not hold...
With a $L^2$-projection approach to conditional expectation, and with $Y_n$ converging in $L^2$ to $Y$, I keep thinking there must be some way of getting this to work... But maybe it just won't.
Thank you for any suggestions!
This is true if $\{\sigma(Y_n)\}$ is an increasing sequence of $\sigma$-fields s.t. $\sigma(Y_n)\nearrow \sigma(Y)$. In this case (see Theorem 4.6.8 on page 247 here), $$ \mathsf{E}[X\mid Y_n]\to \mathsf{E}[X\mid Y]\quad\text{a.s. and in $L^1$}. $$ For example, this holds when $Y_n=2^{-n}\lceil 2^nY\rceil$.