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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 $C_y >0$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1206.6953","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-06-29T06:43:38Z","cross_cats_sorted":[],"title_canon_sha256":"ef18a067e31a08094fc13f0b80ca9a4c0b47dfbc530cd921cbca1b76b2138a4c","abstract_canon_sha256":"05e62c6cb9c6c0cc62ba377636fae6594511fec9e54ab25841012b8261cd2d23"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:52:13.405407Z","signature_b64":"nR8TeSL13aVvpQ7qIha6br86s9wzCEnUdjhpvaK4NG6aEIXniNnBRiW8XwDwOXlng2cel0DTVrSFpCnTJcd6AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6cbbd2692ac775d0d3adc7c3c7e7bffc63adbf30b93189edb8173c8f925dd515","last_reissued_at":"2026-05-18T03:52:13.404580Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:52:13.404580Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1206.6953","created_at":"2026-05-18T03:52:13.404725+00:00"},{"alias_kind":"arxiv_version","alias_value":"1206.6953v1","created_at":"2026-05-18T03:52:13.404725+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.6953","created_at":"2026-05-18T03:52:13.404725+00:00"},{"alias_kind":"pith_short_12","alias_value":"NS55E2JKY525","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_16","alias_value":"NS55E2JKY525BU5N","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_8","alias_value":"NS55E2JK","created_at":"2026-05-18T12:27:16.716162+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NS55E2JKY525BU5NY7B4PZ577R","json":"https://pith.science/pith/NS55E2JKY525BU5NY7B4PZ577R.json","graph_json":"https://pith.science/api/pith-number/NS55E2JKY525BU5NY7B4PZ577R/graph.json","events_json":"https://pith.science/api/pith-number/NS55E2JKY525BU5NY7B4PZ577R/events.json","paper":"https://pith.science/paper/NS55E2JK"},"agent_actions":{"view_html":"https://pith.science/pith/NS55E2JKY525BU5NY7B4PZ577R","download_json":"https://pith.science/pith/NS55E2JKY525BU5NY7B4PZ577R.json","view_paper":"https://pith.science/paper/NS55E2JK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1206.6953&json=true","fetch_graph":"https://pith.science/api/pith-number/NS55E2JKY525BU5NY7B4PZ577R/graph.json","fetch_events":"https://pith.science/api/pith-number/NS55E2JKY525BU5NY7B4PZ577R/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NS55E2JKY525BU5NY7B4PZ577R/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NS55E2JKY525BU5NY7B4PZ577R/action/storage_attestation","attest_author":"https://pith.science/pith/NS55E2JKY525BU5NY7B4PZ577R/action/author_attestation","sign_citation":"https://pith.science/pith/NS55E2JKY525BU5NY7B4PZ577R/action/citation_signature","submit_replication":"https://pith.science/pith/NS55E2JKY525BU5NY7B4PZ577R/action/replication_record"}},"created_at":"2026-05-18T03:52:13.404725+00:00","updated_at":"2026-05-18T03:52:13.404725+00:00"}