Transforming random variables for having good property?

Question

For arbitrary functions $A$ and $B$ and independent random variables $X$ and $Y$, assume that

\begin{align} \Omega&\triangleq \{(x,y): A(x,y)=1\},\\ \Lambda&\triangleq \{x: B(x)=1\}. \end{align} where $x\in\mathcal{X}$ and $y\in\mathcal{Y}$ are realizations of $X$ and $Y$, respectively.

Let we have $$\mathbb{P}_{X,Y}[\Omega\cap\Lambda]\geq 1-\alpha$$.

I am trying to find functions $f$ and $g$ such that $X'=f(X)$ and $Y'=g(Y)$ and $$\mathbb{P}_{X',Y'}[\Omega\cap\Lambda]=1$$ In addition, $|\mathcal{X}'|$ and $|\mathcal{Y}'|$ should be as much as possible near to $|\mathcal{X}|$ and $|\mathcal{Y}|$, respectively.

Note: One good option for $f$ and $g$ could be binning or grouping function. I guess that there exists functions $f,g$ such that $$\mathbb{P}_{X',Y'}[\Omega\cap\Lambda]=1$$ and $|\mathcal{X}'|=(1-h(\alpha))| \mathcal{X}|$ and $|\mathcal{Y}|=(1-l(\alpha))| \mathcal{Y}|$. $h(\alpha)$ and $l(\alpha)$ go to zero as $\alpha$ goes to zero.

Answer 1

No, of course not.

You cannot generally get $X,Y$ to surely satisfy your desired typical events $\Omega$ and $\Lambda$ by transforming them individually. Let alone finding a somehow negligibly-distorting transformation pair that does so.

Take $Bad$ to be the subset of $(\mathcal{X}\times \mathcal{Y}) \backslash(\Omega\cap \Lambda)$ that has nonzero probability. Take $Good_X=\{x:\not\exists y\text{ where }(x,y)\in Bad\}$ and $Good_Y$ similarly. For your transformed variables to surely occur in event $\Omega\cap \Lambda$, then $f$'s range has to be $Good_X$ and $g$'s range has to be $Good_Y$.

Clearly for most distributions $Good_X,\ Good_Y$ are empty.

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There are lots of other deep problems here too, like trying to use binning constructions on an independent single draw of $X$ and $Y$, ambiguity of how $f,g$ should preserve information about the sources they act on (alphabet size doesn't come close to capturing this all).

The only way I can see you might have a chance is when you relax need for sureness to probability $>1-\varepsilon$, where repetitions of the $(X,Y)$ experiment are taken, and when the "repeated" version of the $\Omega \cap \Lambda$ event approaches probability $1$.

But with these changes your problem is already done and no transformation is needed.