{"paper":{"title":"Convergence of Trimmed L\\'evy Processes to Trimmed Stable Random Variables at $0$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Yuguang Fan","submitted_at":"2015-03-18T06:21:58Z","abstract_excerpt":"Let $^{(r,s)}X_t$ be the L\\'evy process $X_t$ with the $r$ largest jumps and $s$ smallest jumps up till time $t$ deleted and let $^{(r)}\\tilde X_t$ be $X_t$ with the $r$ largest jumps in modulus up till time $t$ deleted. We show that $({}^{(r,s)}X_t - a_t)/b_t$ or $({}^{(r)}\\tilde X_t - a_t)/b_t$ converges to a proper nondegenerate nonnormal limit distribution as $t \\downarrow 0$ if and only if $(X_t-a_t)/b_t $ converges as $t \\downarrow 0$ to an $\\alpha$-stable random variable, with $ 0 <\\alpha<2 $, where $a_t$ and $b_t>0$ are non stochastic functions in $t$. Together with the asymptotic norm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.05290","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"}