  The propagation of light in a resonant atomic vapor can \textit{a priori} be
thought of as a multiple scattering process, in which each scattering event
redistributes both the direction and the frequency of the photons.
Particularly, the frequency redistribution may result in L\'evy flights of
photons, directly affecting the transport properties of light in a resonant
atomic vapor and turning this propagation into a superdifusion process. Here,
we report on a Monte-Carlo simulation developed to study the evolution of the
spectrum of the light in a resonant thermal vapor. We observe the gradual
change of the spectrum and its convergence towards a regime of Complete
Frequency Redistribution as the number of scattering events increases. We also
analyse the probability density function of the step length of photons between
emissions and reabsorptions in the vapor, which governs the statistics of the
light diffusion. We observe two different regime in the light transport:
superdiffusive when the vapor is excited near the line center and normal
diffusion for excitation far from the line center. The regime of Complete
Frequency Redistribution is not reached for excitation far from resonance even
after many absorption/reemission cycles due to correlations between emitted and
absorbed frequencies.
