{"paper":{"title":"Limit theorems for Hilbert space-valued linear processes under long range dependence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Marie-Christine D\\\"uker","submitted_at":"2017-01-03T10:31:08Z","abstract_excerpt":"Let $(X_{k})_{k \\in \\mathbb Z }$ be a linear process with values in a separable Hilbert space $\\mathbb{H}$ given by $X_{k} =\\sum_{j=0}^{\\infty} (j+1)^{-N}\\varepsilon_{k-j}$ for each $k \\in \\mathbb Z$, where $N:\\mathbb{H} \\to \\mathbb{H}$ is a bounded, linear normal operator and $(\\varepsilon_{k})_{ k \\in \\mathbb Z }$ is a sequence of independent, identically distributed $\\mathbb{H}$-valued random variables with $E\\varepsilon_{0}=0$ and $E\\| \\varepsilon_{0} \\|^2<\\infty$. We investigate the central and the functional central limit theorem for $(X_{k})_{k \\in \\mathbb Z }$ when the series of operat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.00625","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"}