{"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. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.0612","kind":"arxiv","version":3},"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"}