{"paper":{"title":"On the Limiting Ratio of Current Age to Total Life for Null Recurrent Renewal Processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hermann Thorisson, Jose Blanchet, Peter Glynn","submitted_at":"2015-03-29T00:47:24Z","abstract_excerpt":"If the inter-arrival time distribution of a renewal process is regularly varying with index $\\alpha\\in\\left( 0,1\\right) $ (i.e. the inter-arrival times have infinite mean) and if $A\\left( t\\right) $ is the associated age process at time $t$. Then we show that if $C\\left( t\\right) $ is the length of the current cycle at time $t$, \\[ A\\left( t\\right) /C\\left( t\\right) \\Rightarrow U^{1/\\alpha}, \\] where $U$ is $U\\left( 0,1\\right) $. This extends a classical result in renewal theory in the finite mean case which indicates that the limit is $U\\left( 0,1\\right) $."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.08374","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}