{"paper":{"title":"Functional limit theorems for sums of independent geometric L\\'{e}vy processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Zakhar Kabluchko","submitted_at":"2009-11-21T16:27:31Z","abstract_excerpt":"Let $\\xi_i$, $i\\in \\mathbb {N}$, be independent copies of a L\\'{e}vy process $\\{\\xi(t),t\\geq0\\}$. Motivated by the results obtained previously in the context of the random energy model, we prove functional limit theorems for the process \\[Z_N(t)=\\sum_{i=1}^N\\mathrm{e}^{\\xi_i(s_N+t)}\\] as $N\\to\\infty$, where $s_N$ is a non-negative sequence converging to $+\\infty$. The limiting process depends heavily on the growth rate of the sequence $s_N$. If $s_N$ grows slowly in the sense that $\\liminf_{N\\to\\infty}\\log N/s_N>\\lambda_2$ for some critical value $\\lambda_2>0$, then the limit is an Ornstein--U"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.4139","kind":"arxiv","version":2},"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"}