A class of time-domain procedures for testing that a stationary time-series is Gaussian is presented and analyzed. These tests are based on the deviations of the sample value of finite memory nonlinear transformations of the process from their ensemble averaged counterparts. Asymptotic distributions of these tests are derived under the null hypothesis of Gaussianity and under a class of local and fixed alternatives. Specific tests are then developed, based, respectively, on higher order moments and on the characteristic functions. Practical construction of the test statistics is discussed, with a special emphasis on the estimation of the covariance of the sample statistics, which appears to play a key role in the performance of the tests when dealing with 'small' samples.
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