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With $X$ the number of bulbs on at the terminal time $n$, an even integer, and $\\mu=n/2, \\sigma^2=Var(X)$, we have $$ \\sup_{z \\in \\mathbb{R}} |P(\\frac{X-\\mu}{\\sigma} \\le z)-P(Z \\le z)| \\le \\frac{n}{2\\sigma^2} \\bar{\\Delta}_0 + 1.64 \\frac{n}{\\sigma^3}+ \\frac{2}{\\sigma} $$ where $Z$ is a standard normal random variable, and $$ \\bar{\\Delta}_0 = 1/2\\sqrt{n}} + \\frac{1}{2n} + 1/3"},"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":"1001.0612","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-01-05T01:09:38Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"a0d374609b3063abd4507c17ef8b9b68bd32e9fae90c4dfb6eda0612fd541b98","abstract_canon_sha256":"d4914f5aebc48ada16c1e44604a2f9e20f1f49510f319f1ccba00a239d8df501"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:28:20.414013Z","signature_b64":"tzS1F4XBvhd91aPSxS1MUZxsAClc7gtoxrXlhRFvKdb+7x4dKSNqmR1A2RRzzL3Q4R/pFeIRigugBK6bjHYIDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"99b6b8e025df39cb3cb97098f6ae1d01e6bdbdbc86f2d2859155152e608be847","last_reissued_at":"2026-05-18T04:28:20.413524Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:28:20.413524Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Berry Esseen Theorem for the Lightbulb Process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Haimeng Zhang, Larry Goldstein","submitted_at":"2010-01-05T01:09:38Z","abstract_excerpt":"In the so called lightbulb process, on days $r=1,..., n$, out of $n$ lightbulbs, all initially off, exactly $r$ bulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. 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