{"paper":{"title":"Realizations and Factorizations of Positive Definite Kernels","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.SP","authors_text":"Feng Tian, Palle Jorgensen","submitted_at":"2017-11-09T21:47:36Z","abstract_excerpt":"Given a fixed sigma-finite measure space $\\left(X,\\mathscr{B},\\nu\\right)$, we shall study an associated family of positive definite kernels $K$. Their factorizations will be studied with view to their role as covariance kernels of a variety of stochastic processes. In the interesting cases, the given measure $\\nu$ is infinite, but sigma-finite. We introduce such positive definite kernels $K\\left(\\cdot,\\cdot\\right)$ with the two variables from the subset of the sigma-algebra $\\mathscr{B}$, sets having finite $\\nu$ measure. Our setting and results are motivated by applications. The latter are co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.03614","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"}