Linear and Non-Linear Independence of Random Variable

Question

Is there such thing as non-linearly independent(dependent) variables, and if so, how it is defined?

Also, if the variables are linearly independent, can they be non-linearly dependent?

Thanks in advance)).

Answer 1

Broadly, yes - two random variables may be related in a non-linear way. The simplest example would be where one is directly calculated from the other with a non-linear function, for example $Y = X^2$.

We tend to not say "linearly independent", but we might say that the variables are "linearly uncorrelated" if their correlation coefficient is zero. It is absolutely possible for variables with zero linear correlation to have a non-linear dependence. For example, if $X \sim U(-1, 1)$ (i.e. $X$ is uniformly distributed between -1 and 1) and $Y = X^2$, then $\rho(X, Y) = 0$ but $X$ and $Y$ are definitely not independent because $P(X \leq x, Y \leq y) \neq P(X \leq x)P(Y \leq y)$ for most important values of $x$ and $y$.