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 and a mixing condition. The calculus is long and tedious but the final result is simple and is implemented with a low computational burden. The proposed expression of the test power exhibits good properties on various scenarios; this is illustrated by computer experiments by means of copulas.