 On the ratio of current age to total life for null recurrent renewal processesJohn Angus and Yujia Ding Statistics & Probability Letters, 2020, vol. 162, issue C Abstract: A number of open problems associated with determining the limit distribution of the ratio of current age to total life for a null recurrent renewal process (i.e. where inter-arrival times have infinite mean) are solved. In particular, when the survival function for the inter-arrival times satisfies F̄(t)∼t−αL(t) as t→∞ with L slowly varying and 0≤α≤1, we prove that the limit distribution corresponds to that of U1∕α, where U is uniformly distributed on (0,1), with the limit distribution taken to be degenerate at 0 when α=0. By using direct methods instead of appealing to strong renewal theorems, we are able to prove this result without regard to whether the inter-arrival time distribution is latticed or not, and without extraneous constraints on the renewal function. Keywords: Infinite mean; Regular variation; Convergence in distribution; Renewal theorems (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:eee:stapro:v:162:y:2020:i:c:s0167715220300481 Ordering information: This journal article can be ordered from DOI: 10.1016/j.spl.2020.108745 Access Statistics for this article Statistics & Probability Letters is currently edited by Somnath Datta and Hira L. Koul More articles in Statistics & Probability Letters from Elsevier |
Is your work missing from RePEc? Here is how to contribute.
Questions or problems? Check the EconPapers FAQ or send mail to .
EconPapers is hosted by the
School of Business at Örebro University.