Does one of these two assertions imply the other ?
(1) $X_1, X_2, ..., X_n$ are linearly independent random variables (i.e. $\lambda_1 X_1 + \lambda_2 X_2 + ... + \lambda_n X_n = 0$ => $\lambda_1 =\lambda_2=...=\lambda_n=0$)
(2) $X_1, X_2, ..., X_n$ are independent random variables (stochastically independent)
If not, is there some special cases for which one implication (1=>2 or 2=>1) is true (Gaussian law? etc.)