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We prove the relative asymptotic behavior for the ratio of two sequences of multiple orthogonal polynomials with respect to the Nikishin systems of measures. The first Nikishin system inlMMLBox is such that for each k , sigma k has a constant sign on its compact support inlMMLBox consisting of an interval inlMMLBox , on which inlMMLBox almost everywhere, and a discrete set without accumulation points in inlMMLBox . If inlMMLBox denotes the smallest interval containing inlMMLBox , we assume that Delta k ∩ Delta k +1 =0/ , k =1,..., m −1 . The second Nikishin system inlMMLBox is a perturbation of the first by means of rational functions r k , k =1,..., m , whose zeros and poles lie in inlMMLBox .