When Does Mean Independence Imply Full Independence? A Deep Dive

Definition (continuous random variables X and Y): if the joint density equals the product of the marginal densities, i.e., f_{X,Y}(x,y)=f_X(x)f_Y(y), then X and Y are said to be independent.

Independence is the strongest notion of “no relationship” between random variables. Linear uncorrelatedness is weaker, requiring only Cov(X,Y)=0. Independence implies linear uncorrelatedness, but not the reverse. Between them lies an intermediate notion called “mean independence”.

Definition (conditional expectation): Assume the conditional expectation E[Y|X] exists. If E[Y|X] does not depend on X, we say that Y is mean‑independent of X (Y is mean‑independent of X). Mean independence is not a symmetric relation; Y being mean‑independent of X does not imply X is mean‑independent of Y.

Proposition: Y is mean‑independent of X if and only if E[Y|X]=E[Y].

Proof: (1) Suppose Y is mean‑independent of X, then E[Y|X] is a constant (does not depend on X), hence E[Y|X]=E[Y]. (2) Conversely, if E[Y|X]=E[Y], then the conditional expectation clearly does not depend on X, so Y is mean‑independent of X.

Proposition: If X and Y are independent, then Y is mean‑independent of X, and X is mean‑independent of Y.

Theorem (mean independence implies uncorrelatedness): If Y is mean‑independent of X, or X is mean‑independent of Y, then Cov(Y,X)=0. Proof uses the definition of covariance, the law of iterated expectations, treating constants appropriately, and the linearity of the expectation operator together with the definition of mean independence.

In summary, “independence” ⇒ “mean independence” ⇒ “linear uncorrelatedness”.