Quasi-independence of random variables and a property of the normal and gamma distributions

Random elements ξ,η of a general nature are called quasi-independent, if P(ξ∈A)P(η∈B)>0 implies P(ξ∈A,η∈B)>0. Some properties of quasi-independence are proved. If X1,…,Xn,n⩾3 are independent identically distributed random variables with a distribution function F, quasi-independence of residuals Xi-X¯,Xj-X¯ (holding in case of positive F′(x)=f(x)) is related to characterization of the normal distribution F by the property that for any H with finite E{|H(Xj|} the conditional expectation of H(Xj) given the whole vector of residuals depends only on Xj-X¯. A similar result is proved for the gamma distribution.