The strong renewal theorem

2015/07/27 by Caravenna, Francesco
#60G50 #60K05 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1507.07502

Abstract

We consider real random walks with positive increments (renewal processes) in the domain of attraction of a stable law with index α∈ (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 α> (1)/(2). Understanding when the strong renewal theorem holds for α≤ (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 This paper is superseded by arXiv:1612.07635

Related

• Integral criteria for Strong Renewal Theorems with infinite mean
• The strong renewal theorem with infinite mean via local large deviations
• Strong renewal theorems and local large deviations for multivariate random walks and renewals
• Strong renewal theorem and local limit theorem in the absence of regular\n variation
• On Multivariate Strong Renewal Theorem