I have a very specific question concerning notation I have seen used in proofs in econometric journals.
I want to home in on the difference(s) between $\perp$ versus ${\perp\!\!\!\perp}$.
On the one hand, I have seen the former glyph used to denote orthogonality in linear algebra. On the other hand, the latter glyph is typically used to demonstrate independence of events, such as $A{\perp\!\!\!\perp}B$ (i.e., events A and B are independent) in probability and statistics.
Could the two symbols be used interchangeably?
I appreciate any insight.
The statistical equivalent for random variables to be orthogonal is being uncorrelated.
If $X$ and $Y$ are independent, then they are also uncorrelated, however the converse does not hold. Consider for example a random variable $X\sim N(0,1)$ and compute $$Cov(X,X^2)=E[XX^2]-E[X]E[X^2] = 0.$$ This means that $X$ and $X^2$ are uncorrelated, but $X$ and $X^2$ are clearly not independent. This means that independence is a stronger condition than being uncorrelated. This motivates using the symbol ${\perp\!\!\!\perp}$ to denote independence rather than the "weaker" symbol $\perp$.