I know that in general if the correlation of two random variables is zero, it is always true that two variables are independent, and it is possible to provide some examples which are dependent with zero correlation. However, all examples that I have seen, use nonlinear relation between two variables. Can we say if two random variables have zero correlation will be independent (not using nonlinear relation but linear model)?
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1$\begingroup$ I think your first statement is not correct. Zero correlation does not imply independence. The opposite is true (independence implies zero correlation). $\endgroup$– Kota MoriCommented Dec 16, 2020 at 6:54
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If by that you mean let $X$ be a r.v. with $\mathbb{V}(X) > 0$ and let $Y = aX$ with $a \in \mathbb{R}$ be another random variable, then $0 = \text{Cov}(X,Y) = a \mathbb{V}(X) \iff Y = 0 \ \textrm{a.s.}$
