{"paper":{"title":"Products of Conditional Expectation Operators: Convergence and Divergence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Guolie Lan, Wei Sun, Ze-Chun Hu","submitted_at":"2019-03-10T03:59:44Z","abstract_excerpt":"In this paper, we investigate the convergence of products of conditional expectation operators. We show that if $(\\Omega,\\cal{F},P)$ is a probability space that is not purely atomic, then divergent sequences of products of conditional expectation operators involving 3 or 4 sub-$\\sigma$-fields of $\\cal{F}$ can be constructed for a large class of random variables in $L^2(\\Omega,\\cal{F},P)$. This settles in the negative a long-open conjecture. On the other hand, we show that if $(\\Omega,\\cal{F},P)$ is a purely atomic probability space, then products of conditional expectation operators involving "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.03917","kind":"arxiv","version":3},"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"}