Not necessarily;

Note that sub-independence is defined as follows:

\phi_(X+Y) (t)=\phi_X (t) * \phi_Y (t), where \phi_X (t) stands for the characteristic function of random variable X at t.

Clearly, if X and Y are independent then they are sub-independent; but the converse does not hold necessarily.

If X and Y be sub-independent, then

(1) if cov(X,Y) exists, then X and Y are uncorrelated.

(2) there exists some cases for which X and Y are sub-independent, but X and \alpha*Y are not sub-independent.

Suppose X and Y are two random variables which are not independent, with probability density functions, f(x) and f(y), respectively. Let Z = X + Y. Can we find an example of two probability distributions for X and Y, belonging to the same or different families, such that f(z) = f(x) . f(y)?

Regarding your question, f(z) cannot be equal to f(x).f(y), since the second is a bivariate function of (x,y) but the first is an univariate function of z.

Thank you.