{"paper":{"title":"On the uniform convergence of random series in Skorohod space and representations of c\\`{a}dl\\`{a}g infinitely divisible processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Andreas Basse-O'Connor, Jan Rosi\\'nski","submitted_at":"2011-11-07T19:13:40Z","abstract_excerpt":"Let $X_n$ be independent random elements in the Skorohod space $D([0,1];E)$ of c\\`{a}dl\\`{a}g functions taking values in a separable Banach space $E$. Let $S_n=\\sum_{j=1}^nX_j$. We show that if $S_n$ converges in finite dimensional distributions to a c\\`{a}dl\\`{a}g process, then $S_n+y_n$ converges a.s. pathwise uniformly over $[0,1]$, for some $y_n\\in D([0,1];E)$. This result extends the It\\^{o}-Nisio theorem to the space $D([0,1];E)$, which is surprisingly lacking in the literature even for $E=R$. The main difficulties of dealing with $D([0,1];E)$ in this context are its nonseparability unde"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.1682","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"}