A perturbation of CHSH inequality induced by fluctuations of ensemble distributions

We reconsider the theory of hidden variables under the assumption that the conjecture on the ensemble (experiment run) independence of the distribution of hidden variables (which was indirectly used by J. Bell and his followers) is violated. Ensemble fluctuations imply perturbations of Bell’s inequality and its generalizations. We study (by experimental reasons) CHSH (Clauser, Horne, Shimony, Holt) inequality and obtain its modification. This modified inequality is not in disaccord with the predictions of quantum formalism. The deviation from the standard CHSH inequality depends on the magnitude of ensemble fluctuations. We find these magnitude for fluctuating families of Gaussian distributions. We found that if the dimension of the space of hidden variables is very high, then to obtain a contradiction between the local realism and quantum formalism, we must be sure there is no even negligibly small deviations in probability distributions of hidden variables corresponding to different runs of the experiment (in particular, the efficiency of detectors must be equal to one).