On a central limit theorem in renewal theory

2023/05/22 by Svante Janson, Janson, Svante
Economics, Econometrics and Finance · Mathematics · #60K05 #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2305.13229

openalex publication_date 2023/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Serfozo (2009, Theorem 2.65) gives a useful central limit theorem for processes with regenerative increments. Unfortunately, there is a gap in the proof. We fill this gap, and at the same time we weaken the assumptions. Furthermore, we give conditions for moment convergence in this setting. We give also further results complementing results in Serfozo (2009) on the law of large numbers and estimates for the mean; in particular, we show that there is a gap between conditions for the weak and strong laws of large numbers.

Related

• A Bound on the Rate of Convergence in the Central Limit Theorem for Renewal Processes under Second Moment Conditions
• Results for convergence rates associated with renewal processes
• Limit Theorems for Markov Renewal Processes
• A note on the convergence of renewal and regenerative processes to a Brownian bridge
• Berry-Esseen estimates for regenerative processes under weak moment assumptions