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K. Mahler introduced the concept of perfect systems in the theory ofsimultaneous Hermite-Pad\'{e} approximation of analytic functions. Recently, weproved that Nikishin systems, generated by measures with bounded support andnon-intersecting consecutive supports contained on the real line, are perfect.Here, we prove that they are also perfect when the supports of the generatingmeasures are unbounded or touch at one point. As an application, we give aversion of Stieltjes' theorem in the context of simultaneous Hermite-Pad\'eapproximation.

Nikishin systems are perfect. Case of unbounded and touching supports