{"paper":{"title":"Asymptotic properties of U-processes under long-range dependence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"C\\'eline L\\'evy-Leduc (LTCI), Eric Moulines (LTCI), H\\'el\\`ene Boistard (GREMAQ), Murad S. Taqqu, Valderio A. Reisen","submitted_at":"2009-12-23T16:53:05Z","abstract_excerpt":"Let $(X_i)_{i\\geq 1}$ be a stationary mean-zero Gaussian process with covariances $\\rho(k)=\\PE(X_{1}X_{k+1})$ satisfying: $\\rho(0)=1$ and $\\rho(k)=k^{-D} L(k)$ where $D$ is in $(0,1)$ and $L$ is slowly varying at infinity. Consider the $U$-process $\\{U_n(r),\\; r\\in I\\}$ defined as $$ U_n(r)=\\frac{1}{n(n-1)}\\sum_{1\\leq i\\neq j\\leq n}\\1_{\\{G(X_i,X_j)\\leq r\\}}\\; , $$ where $I$ is an interval included in $\\rset$ and $G$ is a symmetric function. In this paper, we provide central and non-central limit theorems for $U_n$. They are used to derive the asymptotic behavior of the Hodges-Lehmann estimator"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.4688","kind":"arxiv","version":2},"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"}