Model selection for Markov random fields on graphs under a mixing condition

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We propose a global model selection criterion to estimate the graph of conditional dependencies of a random vector. By global criterion, we mean optimizing a function over the set of possible graphs, eliminating the need to estimate individual neighborhoods and subsequently combine them to estimate the graph. We prove the almost sure convergence of the graph estimator. This convergence holds, provided the data is a realization of a multivariate stochastic process that satisfies a polynomial mixing condition. These are the first results to show the consistency of a model selection criterion for Markov random fields on graphs under non-independent data.