  We consider real random walks with positive increments (renewal processes) in
the domain of attraction of a stable law with index $\alpha \in (0,1)$. The
famous local renewal theorem of Garsia and Lamperti, also called strong renewal
theorem, is known to hold in complete generality only for $\alpha >
\frac{1}{2}$. Understanding when the strong renewal theorem holds for $\alpha
\le \frac{1}{2}$ is a long-standing problem, with sufficient conditions given
by Williamson, Doney and Chi. In this paper we give a complete solution,
providing explicit necessary and sufficient conditions (an analogous result has
been independently and simultaneously proved by Doney in arXiv:1507.06790). We
also show that these conditions fail to be sufficient if the random walk is
allowed to take negative values.
  This paper is superseded by arXiv:1612.07635
