Statistics of neutrinos and the double beta decay

We assume that the Pauli exclusion principle is violated for neutrinos, and thus, neutrinos obey at least partly the Bose-Einstein statistics. The parameter sin^2 chi is introduced that characterizes the bosonic (symmetric) fraction of the neutrino wave function. Consequences of the violation of the exclusion principle for the two-neutrino double beta decays are considered. This violation strongly changes the rates of the decays and modifies the energy and angular distributions of the emitted electrons. Pure bosonic neutrinos are excluded by the present data. In the case of partly bosonic (or mixed-statistics) neutrinos the analysis of the existing data allows to put the conservative upper bound sin^2 chi < 0.6. The sensitivity of future measurements of the two-neutrino double beta decay to sin^2 chi is evaluated.