Haar negligibility of positive cones in Banach spaces

The Haar negligibility of the positive cone associated with a basic sequence is discussed in the case of a separable Banach space. In particular, it is shown that, up to equivalence, the canonical basis of $c_0$ is the only normalized subsymmetric unconditional basic sequence whose positive cone is not Haar null, and the only normalized unconditional basic sequence whose positive cone contains a translate of every compact set. It is also proved that an unconditional basic sequence with a non-Haar null positive cone must be $c_0$-saturated in a very strong sense, and that every quotient of the space generated by such a sequence is $c_0$-saturated.