{"paper":{"title":"A Gaussian upper bound for martingale small-ball probabilities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Charles K. Smart, James R. Lee, Yuval Peres","submitted_at":"2014-05-23T07:46:41Z","abstract_excerpt":"Consider a discrete-time martingale $\\{X_t\\}$ taking values in a Hilbert space $\\mathcal H$. We show that if for some $L \\geq 1$, the bounds $\\mathbb{E} \\left[\\|X_{t+1}-X_t\\|_{\\mathcal H}^2 \\mid X_t\\right]=1$ and $\\|X_{t+1}-X_t\\|_{\\mathcal H} \\leq L$ are satisfied for all times $t \\geq 0$, then there is a constant $c = c(L)$ such that for $1 \\leq R \\leq \\sqrt{t}$, \\[\\mathbb{P}(\\|X_t\\|_{\\mathcal H} \\leq R \\mid X_0 = x_0) \\leq c \\frac{R}{\\sqrt{t}} e^{-\\|x_0\\|_{\\mathcal H}^2/(6 L^2 t)}\\,.\\] Following [Lee-Peres, Ann. Probab. 2013], this has applications to diffusive estimates for random walks on "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.5980","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"}