I am reading lecture note (Stanford stat361: https://web.stanford.edu/~swager/stats361.pdf) written by Stefan Wager. At page 23-24 the author states dependent summands become independent after conditioning. Is there someone can explain this formally? In addition, the conditional variance used here also seems weird.
By the way, this note is great since there is only few systematic introduction about the causality. However, it is not rigorous enough with respect to math language. Can someone recommend some other references?