Motivation a special case

A special case of this question is an inequality between the Spearman's rank correlation and the Pearson correlation (Why is the sum of individual Spearman's rho squared less than 1 as opposed to Pearson's r in a synthetic example? ). The expectation of the squared sample Spearman's rank correlation will approach $\rho_{f(X),g(X+Y)}^2$ with $f$ and $g$ the cumulative distribution function of the variables $X$ and $X+Y$.

By trying out several different distributions for $X$ and $Y$ it seems that in all cases the Spearman's rank correlation coefficient is smaller than the Pearson's correlation coefficient.

Thinking about that question it seemed to me intuitive that the correlation between $X$ and $X+Y$ must be a maximum. When we apply a formula, then information gets lost and the correlation should decrease. I imagine that a correlation between $f(X)$ and $g(X+Y)$ can't be larger than a correlation between the original $X$ and $X+Y$.

General question

So I am wondering whether, beyond that example with the Spearman's rank correlation it is true in general for other functions as well:

If $X$ and $Y$ are independent, then the correlation should decrease or at least stay the same when we apply functions to $X$ and $X+Y$. $\rho_{f(X),g(X+Y)}^2 \leq \rho_{X,X+Y}^2$

Is this true?

Is there a simple proof for this?

Motivation a special case

A special case of this question is an inequality between the Spearman's rank correlation and the Pearson correlation (Why is the sum of individual Spearman's rho squared less than 1 as opposed to Pearson's r in a synthetic example? ). The expectation of the squared sample Spearman's rank correlation will approach $\rho_{f(X),g(X+Y)}^2$ with $f$ and $g$ the cumulative distribution function of the variables $X$ and $X+Y$.

By trying out several different distributions for $X$ and $Y$ it seems that in all cases the Spearman's rank correlation coefficient is smaller than the Pearson's correlation coefficient.

Thinking about that question it seemed to me intuitive that the correlation between $X$ and $X+Y$ must be a maximum. When we apply a formula, then information gets lost and the correlation should decrease. I imagine that a correlation between $f(X)$ and $g(X+Y)$ can't be larger than a correlation between the original $X$ and $X+Y$.

General question

So I am wondering whether, beyond that example with the Spearman's rank correlation it is true in general for other functions as well:

If $X$ and $Y$ are independent, then the correlation should decrease or at least stay the same when we apply functions to $X$ and $X+Y$. $\rho_{f(X),g(X+Y)}^2 \leq \rho_{X,X+Y}^2$

Is this true?

Is there a simple proof for this?

Edit: Apparently there can be made many examples relatively easy. Could it also work when $Var(Y) > Var(X)$ and $f$ and $g$ are monotonic functions?

Motivation a special case

A special case of this question is an inequality between the Spearman's rank correlation and the Pearson correlation (Why is the sum of individual Spearman's rho squared less than 1 as opposed to Pearson's r in a synthetic example? ). The expectation of the squared sample Spearman's rank correlation will approach $\rho_{f(X),g(X+Y)}^2$ with $f$ and $g$ the cumulative distribution function of the variables $X$ and $X+Y$.

By trying out several functions it seems that for any variables $X$ and $Y$ the Spearman's rank correlation coefficient is smaller than the Pearson's correlation coefficient.

Thinking about that question it seemed to me intuitive that the correlation between $X$ and $X+Y$ must be a maximum. When we apply a formula, then information gets lost and the correlation should decrease. I imagine that a correlation between $f(X)$ and $g(X+Y)$ can't be larger than a correlation between the original $X$ and $X+Y$.

General question

So I am wondering whether, beyond that example with the Spearman's rank correlation it is true in general for other functions as well:

If $X$ and $Y$ are independent, then the correlation should decrease or at least stay the same when we apply functions to $X$ and $X+Y$. $\rho_{f(X),g(X+Y)}^2 \leq \rho_{X,X+Y}^2$

Is this true?

Is there a simple proof for this?

Motivation a special case

A special case of this question is an inequality between the Spearman's rank correlation and the Pearson correlation (Why is the sum of individual Spearman's rho squared less than 1 as opposed to Pearson's r in a synthetic example? ). The expectation of the squared sample Spearman's rank correlation will approach $\rho_{f(X),g(X+Y)}^2$ with $f$ and $g$ the cumulative distribution function of the variables $X$ and $X+Y$.

By trying out several different distributions for $X$ and $Y$ it seems that in all cases the Spearman's rank correlation coefficient is smaller than the Pearson's correlation coefficient.

Thinking about that question it seemed to me intuitive that the correlation between $X$ and $X+Y$ must be a maximum. When we apply a formula, then information gets lost and the correlation should decrease. I imagine that a correlation between $f(X)$ and $g(X+Y)$ can't be larger than a correlation between the original $X$ and $X+Y$.

General question

So I am wondering whether, beyond that example with the Spearman's rank correlation it is true in general for other functions as well:

If $X$ and $Y$ are independent, then the correlation should decrease or at least stay the same when we apply functions to $X$ and $X+Y$. $\rho_{f(X),g(X+Y)}^2 \leq \rho_{X,X+Y}^2$

Is this true?

Is there a simple proof for this?