{"paper":{"title":"Tightness and Convergence of Trimmed L\\'evy Processes to Normality at Small Times","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Yuguang Fan","submitted_at":"2014-10-19T05:12:42Z","abstract_excerpt":"Let $^{(r,s)}X_t$ be the L\\'evy process $X_t$ with the $r$ largest positive jumps and $s$ smallest negative jumps up till time $t$ deleted and let $^{(r)}\\widetilde X_t$ be $X_t$ with the $r$ largest jumps in modulus up till time $t$ deleted. Let $a_t \\in \\mathbb{R}$ and $b_t>0$ be non-stochastic functions in $t$. We show that the tightness of $({}^{(r,s)}X_t - a_t)/b_t$ or $({}^{(r)}\\widetilde X_t - a_t)/b_t$ at $0$ implies the tightness of all normed ordered jumps, hence the tightness of the untrimmed process $(X_t -a_t)/b_t$ at $0$. We use this to deduce that the trimmed process $({}^{(r,s)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5036","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"}