  We analyze the validity of Bell and Kochen-Specker theorems under local (or
noncontextual) realism but avoiding an assumption of the existence of a joint
probability distribution for incompatible observables. We formulate a realist
model which complies with this requirement. This is obtained by employing
divergent sequences that nevertheless have marginals which are convergent. We
find that under standard reasonable assumptions this possibility does not lead
to a loophole of those theorems, by deriving a short of CHSH inequality valid
for any finite size ensemble. Moreover, we analyze a Hardy's paradox setting
where noncontextual realism imposes the existence of joint probabilities for
incompatible observables.
