The strong renewal theorem

Abstract

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 This paper is superseded by arXiv:1612.07635

31 pages, fixed typos. Superseded by arXiv:1612.07635

References in corpus (2)

• The strong renewal theorem with infinite mean via local large deviations
• Integral criteria for Strong Renewal Theorems with infinite mean

Cited by in corpus (2)

• The strong renewal theorem with infinite mean via local large deviations
• The left tail of renewal measure