Proof that if covariance is zero then there is no linear relationship

I get that a zero covariance doesn´t imply independence, but everybody says that if there is dependence and the covariance is zero then it is a non linear dependence.

People base their interpretation of Pearson’s R in that fact (the closer you are to zero the less linear the relationship is).

Is there a formal proof to that?

I tried to do it by myself but i couldn’t. The proposition i think encapsulates the idea is the following:

If $cov(X,Y)\ne0$ then there exists a Z such that $cov(X,Z)=0$ and $E[Y|X]=bX+E[Z|X]$

Answer

Here is a proof of the mathematical statement at the end of your question: we can find a $Z$ which is uncorrelated to $X$ and satisfies
$$
\mathbb{E}(Y|X) = b X + \mathbb{E}(Z|X)
$$

by assuming $Z = Y – bX$, and then choosing the $b$ which makes $\mathrm{Cov}(X, Z) = 0$ true. For this $b$ we have
$$
0 = \mathrm{Cov}(X, Z) = \mathrm{Cov}(X, Y – bX) = \mathrm{Cov}(X, Y) – b \mathrm{Var}(X),
$$

and thus
$$
b = \frac{\mathrm{Cov}(X, Y)}{\mathrm{Var}(X)}.
$$

(Note that the same $b$ is found as the slope of the linear regression line.) We have $b = 0$, if and only if $\mathrm{Cov}(X,Y) = 0$.

Attribution
Source : Link , Question Author : Gilbert Ibanez , Answer Author : jochen

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