On Sums of Random Variables and Independence
Bernhard K. Flury · The American Statistician · 1986
Suppose that the density of the sum of two random variables X and Y is given by the convolution of the two marginal densities. Although this condition is stronger than uncorrelatedness of X and Y, it does not imply stochastic independence, as is shown by three examples. A situation for which this fact may be relevant occurs in the construction of chi-squared tests for nested hypotheses.