{"paper":{"title":"Phase transitions for a unidirectional elephant random walk with a power law memory II: Some sharper estimates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hideki Tanemura, Masato Takei, Rahul Roy","submitted_at":"2025-04-01T09:23:22Z","abstract_excerpt":"We continue our study of the unidirectional elephant random walk (uERW) initiated in {\\it {Electron. Commun. Probab.}} ({\\bf 29} 2024, article no. 78). In this paper we obtain definitive results when the memory exponent $\\beta\\in (-1, p/(1-p))$. In particular using a coupling argument we obtain the exact asymptotic rate of growth of $S_n$, the location of the uERW at time $n$, for the case $\\beta\\in (-1, 0] $. Also, for the case $\\beta\\in (0, p/(1-p))$ we show that $P(S_n \\to \\infty) \\in (0,1)$ and conditional on $\\{S_n \\to \\infty\\}$ we obtain the exact asymptotic rate of growth of $S_n$. In a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.00566","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2504.00566/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}