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In this work we investigate an alternative to the stochastic Gaussian Maximum Likelihood (ML) method that deals with sub-Gaussian signals. The proposed system is one where the sources are stochastic and Gaussian, and the transfer medium is varying in a highly impulsive manner, introducing the sub-Gaussian nature at the receiver. Alternatively, the impulsive transformation to the signals can be viewed as part of the source model, creating a multivariate source signal whose components cannot be independent and is of impulsiveness equal to the one of the Cauchy distribution. The Levy α-stable distribution, of characteristic exponent 0.5 and index of symmetry 1, is used together with the multivariate Gaussian density to model the signal, and the resulting probability density function is derived. Based on this density, the stochastic ML estimator is formulated. A separable solution of the estimator is given, and simulations demonstrating the performance gains relative to the Gaussian-based ML estimator are provided.