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This paper investigates the evolution of kurtosis of the input Gaussian amplified spontaneous emission (ASE) noise in a nonlinear fiber with negligible dispersion. The nonlinear Schrodinger equation (NLSE) describing propagation in optical fibers is simplified such that the fiber represents a zero memory nonlinear (ZMNL) system, and this approximation allows the development of analytical formulas for the statistical moments of the output noise. It is possible to calculate moments of all integer orders and the explicit expressions for the first four moments are given. The investigations show that the ASE noise does not preserve its Gaussian character when Kerr nonlinearity is significant. This observation proves that the common assumption of the Gaussian output ASE is not necessarily valid. Numerical simulations are provided to support the derivation. Kurtosis deviating significantly from the value typical for Gaussian noise is also an indicator that BER calculation in the coherent systems based on the assumption that ASE is Gaussian is likely to be inaccurate.