Most normality tests in the literature are performed for scalar and independent samples. Thus, they become unreliable when applied to colored processes, hampering their use in realistic scenarios. We focus on Mardia's multivariate kurtosis, derive closed-form expressions of its asymptotic distribution for statistically dependent samples, under the null hypothesis of normality. Included experiments illustrate, by means of copulas, that it does not suffice to test a one-dimensional marginal to conclude normality. The proposed test also exhibits good properties on other typical scenarios, such as the detection of a non-Gaussian process in the presence of an additive Gaussian noise.