Estimating the Renewal Function When the Second Moment Is Infinite

Abstract

We explore a large sample based method for determining the profile of the renewal function when the inter-renewal period does not have a finite variance. Specifically, we construct an empirical estimator and, based on it, develop confidence bands for the renewal and related functions. The estimator's margin of error can be made as small as desired by choosing a sufficiently large sample size and also by considering the renewal function for sufficiently large values of its argument. For small and moderate values of the argument, we suggest using estimators available in the literature and, hence, begin with an extensive literature review on the topic.

Keywords:

• Confidence band
• Consistency
• Empirical process
• Limit distribution
• Renewal function
• Stable law

Mathematics Subject Classification:

• 60K05
• 60E07
• 62G15
• 62G20

ACKNOWLEDGMENTS

This work is partially supported by the Marsden Fund, which is administered by the Royal Society of New Zealand, and an NSERC grant at the University of Western Ontario.

Sincere thanks are due to the Editor, an associate editor, and a referee whose suggestions and comments have helped us to revise the article.