{"paper":{"title":"The functional Breuer-Major theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"David Nualart, Ivan Nourdin","submitted_at":"2018-08-07T14:05:20Z","abstract_excerpt":"Let $X=\\{ X_n\\}_{n\\in \\mathbb{Z}}$ be zero-mean stationary Gaussian sequence of random variables with covariance function $\\rho$ satisfying $\\rho(0)=1$. Let $\\varphi:\\mathbb{R}\\to\\mathbb{R}$ be a function such that $E[\\varphi(X_0)^2]<\\infty$ and assume that $\\varphi$ has Hermite rank $d \\geq 1$. The celebrated Breuer-Major theorem asserts that, if $\\sum_{r\\in\\mathbb{Z}} |\\rho(r)|^d<\\infty$ then the finite dimensional distributions of $\\frac1{\\sqrt{n}}\\sum_{i=0}^{\\lfloor n\\cdot\\rfloor-1} \\varphi(X_i)$ converge to those of $\\sigma\\,W$, where $W$ is a standard Brownian motion and $\\sigma$ is some"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.02378","kind":"arxiv","version":1},"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"}