  The correlation of phase fluctuations in any type of oscillator fundamentally
defines its spectral shape. However, in nonlinear oscillators, such as spin
torque nano oscillators, the frequency spectrum can become particularly
complex. This is specifically true when not only considering thermal but also
colored $1/f$ flicker noise processes, which are crucial in the context of the
oscillator's long term stability. In this study, we address the frequency
spectrum of spin torque oscillators in the regime of large-amplitude steady
oscillations experimentally and as well theoretically. We particularly take
both thermal and flicker noise into account. We perform a series of
measurements of the phase noise and the spectrum on spin torque vortex
oscillators, notably varying the measurement time duration. Furthermore, we
develop the modelling of thermal and flicker noise in Thiele equation based
simulations. We also derive the complete phase variance in the framework of the
nonlinear auto-oscillator theory and deduce the actual frequency spectrum. We
investigate its dependence on the measurement time duration and compare with
the experimental results. Long term stability is important in several of the
recent applicative developments of spin torque oscillators. This study brings
some insights on how to better address this issue.
