First, we recall a concept which is called sub-independence. This concept is stronger than that of uncorrelatedness but a lot weaker than independence. The concept of sub-independence, unlike that of uncorrelatedness, does not depend on the existence of any moments. Then, we consider a particular bivariate mixture to construct a pair (X, Y), which is sub-independent but not independent.
Ahsanullah, M. and Hamedani, G. G., Some characterizations of normal distribution, CSAB, 37 (1988), no. 145–146, pp. 95–99. MR 0964309 (90f:62038)
Hamedani G. G., 'Some characterizations of normal distribution' (1988) 37CSAB: 95-99.
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Hamedani, G. G. and Tata, M. N., On the determination of the bivariate normal from distributions of linear combinations of the variables, Amer. Math. Monthly, 82 (1975), no. 9, pp. 913–915. MR 0383643 (52#4523
Tata M. N., 'On the determination of the bivariate normal from distributions of linear combinations of the variables' (1975) 82Amer. Math. Monthly: 913-915.
Tata M. N.On the determination of the bivariate normal from distributions of linear combinations of the variablesAmer. Math. Monthly197582913915)| false