{"paper":{"title":"Upper functions for $L_p$-norm of gaussian random fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"O. Lepski","submitted_at":"2013-11-20T10:07:02Z","abstract_excerpt":"In this paper we are interested in finding upper functions for a collection of random variables $\\big\\{\\big\\|\\xi_{\\vec{h}}\\big\\|_p, \\vec{h}\\in\\mathrm{H}\\big\\}, 1\\leq p<\\infty$. Here $\\xi_{\\vec{h}}(x), x\\in(-b,b)^d, d\\geq 1$ is a kernel-type gaussian random field and $\\|\\cdot\\|_p$ stands for $L_p$-norm on $(-b,b)^d$. The set $\\mathrm{H}$ consists of $d$-variate vector-functions defined on $(-b,b)^d$ and taking values in some countable net in $R^d_+$. We seek a non-random family $\\left\\{\\Psi_\\alpha\\big(\\vec{h}\\big),\\;\\;\\vec{h}\\in\\mathrm{H}\\right\\}$ such that $ E\\big\\{\\sup_{\\vec{h}\\in\\mathrm{H}}\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.4996","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"}