  Polynomials $Q_n(z)$, $n=0,1,\ldots,$ that are multi-orthogonal with respect
to a Nikishin system of $p\geq 1$ compactly supported measures over the
star-like set of $p+1$ rays $S_+:=\{z\in \mathbb{C}: z^{p+1}\geq 0 \}$ are
investigated. We prove that the Nikishin system is normal, that the polynomials
satisfy a three-term recurrence relation of order $p+1$ of the form $z
Q_{n}(z)=Q_{n+1}(z)+a_{n}\,Q_{n-p}(z)$ with $a_n>0$ for all $n\geq p$, and that
the nonzero roots of $Q_n$ are all simple and located in $S_+$. Under the
assumption of regularity (in the sense of Stahl and Totik) of the measures
generating the Nikishin system, we describe the asymptotic zero distribution
and weak behavior of the polynomials $Q_n$ in terms of a vector equilibrium
problem for logarithmic potentials. Under the same regularity assumptions, a
theorem on the convergence of the Hermite-Pad\'e approximants to the Nikishin
system of Cauchy transforms is proven.
