  Pad\'e approximation has two natural extensions to vector rational
approximation through the so called type I and type II Hermite-Pad\'e
approximants. The convergence properties of type II Hermite-Pad\'e approximants
have been studied. For such approximants Markov and Stieltjes type theorems are
available. To the present, such results have not been obtained for type I
approximants. In this paper, we provide Markov and Stieltjes type theorems on
the convergence of type I Hermite-Pad\'e approximants for Nikishin systems of
functions.
