{"paper":{"title":"Small Deviations in $L_2$-norm for Gaussian Dependent Sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexander Nazarov, Mikhail Lifshits, Seok Young Hong","submitted_at":"2015-11-17T12:09:03Z","abstract_excerpt":"Let $U=(U_k)_{k\\in\\mathbb{Z}}$ be a centered Gaussian stationary sequence satisfying some minor regularity condition. We study the asymptotic behavior of its weighted $\\ell_2$-norm small deviation probabilities. It is shown that \\[ \\ln \\mathbb{P}\\left( \\sum_{k\\in\\mathbb{Z}} d_k^2 U_k^2 \\leq \\varepsilon^2\\right) \\sim - M \\varepsilon^{-\\frac{2}{2p-1}}, \\qquad \\textrm{ as } \\varepsilon\\to 0, \\] whenever \\[ d_k\\sim d_{\\pm} |k|^{-p}\\quad \\textrm{for some } p>\\frac{1}{2} \\, , \\quad k\\to \\pm\\infty, \\] using the arguments based on the spectral theory of pseudo-differential operators by M. Birman and M"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.05370","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"}