A martingale view of Blackwell's renewal theorem and its extensions to a general counting process. (June 2019)

• Record Type: Journal Article
• Authors: Daley, Daryl J.; Miyazawa, Masakiyo
• Is Part Of: Journal of applied probability. Volume 56:Number 2(2019)
• Page Start: 602
• Page End: 623
• Publication Date: 2019-06
• Subjects: Renewal process, -- semimartingale, -- Blackwell's renewal theorem, -- key renewal theorem, -- Wald's identity, -- stationary intervals, -- Palm distribution, -- modulated renewal process, -- Smith's asymptotic formulae; Primary 60K25, -- Secondary 60J27, -- 60K37; 519.2
• DOI: 10.1017/jpr.2019.35
• Languages: English
• ISSNs: 0021-9002

Abstract

Abstract: Martingales constitute a basic tool in stochastic analysis; this paper considers their application to counting processes. We use this tool to revisit a renewal theorem and give extensions for various counting processes. We first consider a renewal process as a pilot example, deriving a new semimartingale representation that differs from the standard decomposition via the stochastic intensity function. We then revisit Blackwell's renewal theorem, its refinements and extensions. Based on these observations, we extend the semimartingale representation to a general counting process, and give conditions under which asymptotic behaviour similar to Blackwell's renewal theorem holds.