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In this paper, we present a theoretical and experimental investigation of shot noise in metal-oxide-semiconductor (MOS) capacitors. We show that stress-induced leakage currents exhibit suppressed shot noise with respect to the "full" power-spectral density S=2qI associated to a purely Poissonian process, which is measured in the tunneling current through a fresh oxide. Experimental results on MOS capacitors with 6 and 10 nm oxides are presented. We present a model of stress-induced leakage currents (SILCs) based on trap-assisted tunneling that takes into account elastic and inelastic transitions, and is able to reproduce the relevant physics. Numerical simulations based on the proposed model are presented and exhibit good agreement with the experiments, given the lack of information on the nature of traps.