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We study the formation of extreme events in incoherent systems described byenvelope equations, such as the Nonliner Schr\"odinger equation. We derive anidentity that relates the evolution of the kurtosis (a measure of the relevanceof the tails in a probability density function) of the wave amplitude to therate of change of the width of the Fourier spectrum of the wave field. Theresult is exact for all dispersive systems characterized by a nonlinear term ofthe form of the one contained in the Nonlinear Schr\"odinger equation.Numerical simulations are also performed to confirm our findings. Our worksheds some light on the origin of rogue waves in incoherent dispersivenonlinear media ruled by local cubic nonlinearity.

Origin of heavy tail statistics in equations of the Nonlinear Schr\"odinger type: an exact result