{"paper":{"title":"Return Probabilities for the Reflected Random Walk on $\\mathbb N_0$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Marc Peign\\'e (LMPT), Rim Essifi (LMPT)","submitted_at":"2012-06-29T06:43:38Z","abstract_excerpt":"Let $(Y_n)$ be a sequence of i.i.d. $\\mathbb Z$-valued random variables with law $\\mu$. The reflected random walk $(X_n)$ is defined recursively by $X_0=x \\in \\mathbb N_0, X_{n+1}=|X_n+Y_{n+1}|$. Under mild hypotheses on the law $\\mu$, it is proved that, for any $ y \\in \\mathbb N_0$, as $n \\to +\\infty$, one gets $\\mathbb P_x[X_n=y]\\sim C_{x, y} R^{-n} n^{-3/2}$ when $\\sum_{k\\in \\mathbb Z} k\\mu(k) >0$ and $\\mathbb P_x[X_n=y]\\sim C_{y} n^{-1/2}$ when $\\sum_{k\\in \\mathbb Z} k\\mu(k) =0$, for some constants $R, C_{x, y}$ and $C_y >0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.6953","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"}