{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:VREWYWM3Q3JAGFH2NWJBOA6W7X","short_pith_number":"pith:VREWYWM3","schema_version":"1.0","canonical_sha256":"ac496c599b86d20314fa6d921703d6fdf3bc9b5c398d186c2d6faf5a63954343","source":{"kind":"arxiv","id":"0907.2261","version":2},"attestation_state":"computed","paper":{"title":"Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Mariusz Mirek","submitted_at":"2009-07-13T22:51:31Z","abstract_excerpt":"We consider the Markov chain $\\{X_n^x\\}_{n=0}^\\infty$ on $\\R^d$ defined by the stochastic recursion $X_{n}^{x}=\\p_{\\theta_{n}}(X_{n-1}^{x})$, starting at $x\\in\\R^d$, where $\\theta_{1}, \\theta_{2},...$ are i.i.d. random variables taking their values in a metric space $(\\Theta, \\mathfrak{r}),$ and $\\p_{\\theta_{n}}:\\R^d\\mapsto\\R^d$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure $\\nu$. Under appropriate assumptions on $\\p_{\\theta_n}$, we will show that the measure $\\nu$ has a heavy tail with the exponent $\\alpha>0$ i.e. $\\nu(\\{x\\in\\R^d: |x|>t\\})\\asymp t^{-\\alpha}$"},"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":"0907.2261","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-07-13T22:51:31Z","cross_cats_sorted":[],"title_canon_sha256":"d0b12e4a0a4c7b5caaec3cea45656a698c291d95ac0ceebcd74ef4adfd62b93c","abstract_canon_sha256":"a3459d9051ce5d271c072274ec9201166d61837519e33c4a68ffd982d8db0d22"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:36:51.343460Z","signature_b64":"GQGo54X+xAE/WK/SZKUsX3gOfGb7Y2vDlUcuHIm4kNiMSVnJdOlW8nRZ5OA2OGduqqS98Zr6U18li1WN4dT8Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac496c599b86d20314fa6d921703d6fdf3bc9b5c398d186c2d6faf5a63954343","last_reissued_at":"2026-05-18T04:36:51.343045Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:36:51.343045Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Mariusz Mirek","submitted_at":"2009-07-13T22:51:31Z","abstract_excerpt":"We consider the Markov chain $\\{X_n^x\\}_{n=0}^\\infty$ on $\\R^d$ defined by the stochastic recursion $X_{n}^{x}=\\p_{\\theta_{n}}(X_{n-1}^{x})$, starting at $x\\in\\R^d$, where $\\theta_{1}, \\theta_{2},...$ are i.i.d. random variables taking their values in a metric space $(\\Theta, \\mathfrak{r}),$ and $\\p_{\\theta_{n}}:\\R^d\\mapsto\\R^d$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure $\\nu$. Under appropriate assumptions on $\\p_{\\theta_n}$, we will show that the measure $\\nu$ has a heavy tail with the exponent $\\alpha>0$ i.e. $\\nu(\\{x\\in\\R^d: |x|>t\\})\\asymp t^{-\\alpha}$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.2261","kind":"arxiv","version":2},"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":"0907.2261","created_at":"2026-05-18T04:36:51.343102+00:00"},{"alias_kind":"arxiv_version","alias_value":"0907.2261v2","created_at":"2026-05-18T04:36:51.343102+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.2261","created_at":"2026-05-18T04:36:51.343102+00:00"},{"alias_kind":"pith_short_12","alias_value":"VREWYWM3Q3JA","created_at":"2026-05-18T12:26:02.257875+00:00"},{"alias_kind":"pith_short_16","alias_value":"VREWYWM3Q3JAGFH2","created_at":"2026-05-18T12:26:02.257875+00:00"},{"alias_kind":"pith_short_8","alias_value":"VREWYWM3","created_at":"2026-05-18T12:26:02.257875+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/VREWYWM3Q3JAGFH2NWJBOA6W7X","json":"https://pith.science/pith/VREWYWM3Q3JAGFH2NWJBOA6W7X.json","graph_json":"https://pith.science/api/pith-number/VREWYWM3Q3JAGFH2NWJBOA6W7X/graph.json","events_json":"https://pith.science/api/pith-number/VREWYWM3Q3JAGFH2NWJBOA6W7X/events.json","paper":"https://pith.science/paper/VREWYWM3"},"agent_actions":{"view_html":"https://pith.science/pith/VREWYWM3Q3JAGFH2NWJBOA6W7X","download_json":"https://pith.science/pith/VREWYWM3Q3JAGFH2NWJBOA6W7X.json","view_paper":"https://pith.science/paper/VREWYWM3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0907.2261&json=true","fetch_graph":"https://pith.science/api/pith-number/VREWYWM3Q3JAGFH2NWJBOA6W7X/graph.json","fetch_events":"https://pith.science/api/pith-number/VREWYWM3Q3JAGFH2NWJBOA6W7X/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VREWYWM3Q3JAGFH2NWJBOA6W7X/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VREWYWM3Q3JAGFH2NWJBOA6W7X/action/storage_attestation","attest_author":"https://pith.science/pith/VREWYWM3Q3JAGFH2NWJBOA6W7X/action/author_attestation","sign_citation":"https://pith.science/pith/VREWYWM3Q3JAGFH2NWJBOA6W7X/action/citation_signature","submit_replication":"https://pith.science/pith/VREWYWM3Q3JAGFH2NWJBOA6W7X/action/replication_record"}},"created_at":"2026-05-18T04:36:51.343102+00:00","updated_at":"2026-05-18T04:36:51.343102+00:00"}