  We examine the general question of statistical changes experienced by
ensembles of nonlinear random waves propagating in systems ruled by integrable
equations. In our study that enters within the framework of integrable
turbulence, we specifically focus on optical fiber systems accurately described
by the integrable one-dimensional nonlinear Schr\"odinger equation. We consider
random complex fields having a gaussian statistics and an infinite extension at
initial stage. We use numerical simulations with periodic boundary conditions
and optical fiber experiments to investigate spectral and statistical changes
experienced by nonlinear waves in focusing and in defocusing propagation
regimes. As a result of nonlinear propagation, the power spectrum of the random
wave broadens and takes exponential wings both in focusing and in defocusing
regimes. Heavy-tailed deviations from gaussian statistics are observed in
focusing regime while low-tailed deviations from gaussian statistics are
observed in defocusing regime. After some transient evolution, the wave system
is found to exhibit a statistically stationary state in which neither the
probability density function of the wave field nor the spectrum change with the
evolution variable. Separating fluctuations of small scale from fluctuations of
large scale both in focusing and defocusing regime, we reveal the phenomenon of
intermittency; i.e., small scales are characterized by large heavy-tailed
deviations from Gaussian statistics, while the large ones are almost Gaussian.
