{"paper":{"title":"On Integration Methods Based on Scrambled Nets of Arbitrary Size","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"stat.CO","authors_text":"Mathieu Gerber","submitted_at":"2014-08-12T16:46:49Z","abstract_excerpt":"We consider the problem of evaluating $I(\\varphi):=\\int_{[0,1)^s}\\varphi(x) dx$ for a function $\\varphi \\in L^2[0,1)^{s}$. In situations where $I(\\varphi)$ can be approximated by an estimate of the form $N^{-1}\\sum_{n=0}^{N-1}\\varphi(x^n)$, with $\\{x^n\\}_{n=0}^{N-1}$ a point set in $[0,1)^s$, it is now well known that the $O_P(N^{-1/2})$ Monte Carlo convergence rate can be improved by taking for $\\{x^n\\}_{n=0}^{N-1}$ the first $N=\\lambda b^m$ points, $\\lambda\\in\\{1,\\dots,b-1\\}$, of a scrambled $(t,s)$-sequence in base $b\\geq 2$. In this paper we derive a bound for the variance of scrambled net"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.2773","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"}