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I'm guessing that you're trying to formalize a notion similar to this: $${}$$ "$X_1$ and $X_2$ are like $X$, but with different types of noise. $Z$ is a mixing of $X_1$ and $X_2$. There is also some process $Y(\cdot)$ that acts on RVs that live in $\mathcal{X}$. Can I always mix $Z$ so that I learn less about $Y(Z)$ from $Z$ than I learn about $Y(X)$ from $X$?". $${}$$ The answer seems like yes, but what you've written isn't quite the right formalization yet. In particular you need to more closely specify the nature of $X_1$ and $X_2$ (and more cleanly specify the entire problem).
What's more, as you've written it, the LHS $D(P_{Y|Z}|P_{Y_2})$ is going to be a random variable depending on $Z$ and the RHS is a RV depending on $X$. I'm not sure that this is what you are really after.
Math_Y, your definitions are too lax. Nothing about X1, X2 enforces that Z resemble or preserves information about X in any way. This leaves the two sides of your desired inequality unrelated.
I guess I could have meant something by "with high probability"... it could have been that there was a sequence of $S_n$ put together so that the quotient only goes to 1 with probability $f(\alpha,\varepsilon)< 1$.
An argument I believe can be made formal: Your absolute value is going to be the difference of the magnitude of the ones that come out positive from the magnitude of the ones that come out non-positive, or vice-versa. but no matter the selection for which come out positive and which come out negative, the resulting difference has undefined or infinite expected value. from here you can get your limsup to infinity with a large numbers law
