  We prove relative asymptotic for the ratio of two sequences of multiple
orthogonal polynomials with respect to Nikishin system of measures. The first
Nikishin system ${\mathcal{N}}(\sigma_1,...,\sigma_m)$ is such that for each
$k$, $\sigma_k$ has constant sign on its compact support $\supp {\sigma_k}
\subset \mathbb{R}$ consisting of an interval $\widetilde{\Delta}_k$, on which
$|\sigma_k^{\prime}| > 0$ almost everywhere, and a discrete set without
accumulation points in $\mathbb{R} \setminus \widetilde{\Delta}_k$. If
${Co}(\supp {\sigma_k}) = \Delta_k$ denotes the smallest interval containing
$\supp {\sigma_k}$, we assume that $\Delta_k \cap \Delta_{k+1} = \emptyset$,
$k=1,...,m-1$. The second Nikishin system
${\mathcal{N}}(r_1\sigma_1,...,r_m\sigma_m)$ is a perturbation of the first by
means of rational functions $r_k$, $k=1,...,m,$ whose zeros and poles lie in
$\mathbb{C} \setminus \cup_{k=1}^m \Delta_k$.
