Does there exist any general rule describing how the covraince between two dependent and unspecified stochastic variables, X and Y, affects the variance of their product disitrbution (XY)?
For simplicity, assume that E[X] and E[Y] are both positive. Additionally, assume existence of all relevant moments.
By the definition of a covaraince, it is straightforward to show that $$ E[XY]= E[X]E[Y] + \text{Cov}[X,Y] $$ Thus, a negative covariance reduces the expected value of the product disitrbution.
However, the variance seems less trivial. According to Borhrnstedt and Golberger (1969), the variance can be expressed as $$ V[pq] = E^2[X] V[Y] + E^2[Y] V[X] + 2E[X] E[Y]\text{Cov}[X,Y]-\text{Cov}^2(X,Y) \\ + E[(\Delta X)^2(\Delta Y)^2] + 2E[X] E[(\Delta X)(\Delta Y)^2] + 2E[Y] E[(\Delta X)^2(\Delta Y)] $$ with $\Delta X = X-E[X]$ and $\Delta Y = Y-E[Y]$.
The effect of the covariance between X and Y on the variance of the product disitrbution (XY) still seems unclear to me, as the last four terms can be expanded to include Cov[X,Y] in addition to cross-moments. This issue can be solved by asusming specific disitrbutions, but I would like a more general result for two unspecified disitrbutions.