We study spin-torque induced oscillations of MgO magnetic tunnel junctions in the time domain. By using the Hilbert transform on the time traces, we obtain for the first time a direct experimental measure of the coupling between the power and the phase fluctuations. We deduce the power restoration rate and we obtain low values for the coupling strength, which is consistent with the weak frequency dependence on the applied voltage. An electrical current flowing through a ferromagnetic layer is spin-polarized and can exert a torque on a second thin ferromagnetic layer's magnetization, as predicted by Slonczewski 1 and Berger. 2 In certain geometries and even for zero applied field, a steady state precession of the magneti-zation is obtained above a certain threshold when the torque applied by the injected current is large enough to compensate the natural Gilbert damping. In magnetic tunnel junctions ͑MTJ͒, the sustained precession of magnetization is con-verted into microwave voltage signal by the tunnel magne-toresistance effect. It is known that such spin-torque induced nano-oscillator ͑STO͒ is nonlinear, 3,4 i.e., its frequency de-pends on the oscillation power p.