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The purpose of the paper is to find sufficient conditions that guarantee the exponential convergence of smoothing, filtering and predictive probabilities: $$\\sup_{n\\geq t}\\|P(Y_{t:\\infty}\\in \\cdot|X_{l:n})-P(Y_{t:\\infty}\\in \\cdot|X_{s:n}) \\|_{\\rm TV} \\leq K_s \\alpha^{t}, \\quad \\mbox{a.s.}$$ Here $t\\geq s\\geq l\\geq 1$, $K_s$ is $\\sigm"},"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":"2103.05474","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2021-03-09T15:05:30Z","cross_cats_sorted":[],"title_canon_sha256":"fb66ec922fb40c29907fd6c4d54088caf94f2c758598fc5bb8d2754b7e7f30e0","abstract_canon_sha256":"06eeb276bacd974f0e88d37a36e21b8bbd8aac215d9776264beabda581988c79"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:21:26.317405Z","signature_b64":"JMB5XahEAVv1SJT+PLt9Krd9I1CIQwXf0yzao5KauKDjveHgbuucuPfg2X3Se05/WkZ9RV+1cuf4eZthspyZAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"703b310d9afaf12f2362ab1a3b8f4f763b94bb9df8c41da8a5874cf0f3f4f1e7","last_reissued_at":"2026-07-05T02:21:26.316971Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:21:26.316971Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exponential forgetting of smoothing distributions for pairwise Markov models","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Joonas Sova, J\\\"uri Lember","submitted_at":"2021-03-09T15:05:30Z","abstract_excerpt":"We consider a bivariate Markov chain $Z=\\{Z_k\\}_{k \\geq 1}=\\{(X_k,Y_k)\\}_{k \\geq 1}$ taking values on product space ${\\cal Z}={\\cal X} \\times{ \\cal Y}$, where ${\\cal X}$ is possibly uncountable space and ${\\cal Y}=\\{1,\\ldots, |{\\cal Y}|\\}$ is a finite state-space. 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