We investigate the ratio asymptotic behavior of the sequence $(Q_{ n})_{ n=0}^{ \infty}$ of multiple orthogonal polynomials associated with a Nikishin system of $p\geq 1$ measures that are compactly supported on the star-like set of $p+1$ rays $S_{ +}=\{ z\in\mathbb{ C}: z^{ p+1}\geq 0\}$. The main algebraic property of these polynomials is that they satisfy a three-term recurrence relation of the form $zQ_{ n}(z)=Q_{ n+1}(z)+a_{ n} Q_{ n-p}(z)$ with $a_{ n}>0$ for all $n\geq p$. Under a Rakhmanov-type condition on the measures generating the Nikishin system, we prove that the sequence of ratios $Q_{ n+1}(z)/Q_{ n}(z)$ and the sequence $a_{ n}$ of recurrence coefficients are limit periodic with period $p(p+1)$. Our results complement some results obtained by the first author and Mi\~{ n}a-D\'{ i}az in a recent paper in which algebraic properties and weak asymptotics of these polynomials were investigated. Our results also extend some results obtained by the first author in the case $p=2$.
Categories: math.CA math.SP