Covariance, Correlation and Independence

Statements

Theorem. Let $X_1$ and $X_2$ be random variables with covariance $\text{Cov}(X_1,X_2)$ and correlation $\text{Corr}(X_1,X_2)$. If $X_1$ and $X_2$ are independent random variables, then $\text{Cov}(X_1,X_2)=0$ and $X_1$ and $X_2$ are uncorrelated. Conversely, if $\text{Cov}(X_1,X_2)=0$ and hence $X_1$ and $X_1$ are uncorrelated, then $X_1$ and $X_1$ are not necessarily independent.

References

Ost20, Theorem 6.4.2