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We analyze the validity of Bell and Kochen-Specker theorems under local (ornoncontextual) realism but avoiding an assumption of the existence of a jointprobability distribution for incompatible observables. We formulate a realistmodel which complies with this requirement. This is obtained by employingdivergent sequences that nevertheless have marginals which are convergent. Wefind that under standard reasonable assumptions this possibility does not leadto a loophole of those theorems, by deriving a short of CHSH inequality validfor any finite size ensemble. Moreover, we analyze a Hardy's paradox settingwhere noncontextual realism imposes the existence of joint probabilities forincompatible observables.

On the role of joint probability distributions of incompatible observables in Bell and Kochen-Specker Theorems