{"paper":{"title":"Asymptotics of random processes with immigration I: scaling limits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexander Iksanov, Alexander Marynych, Matthias Meiners","submitted_at":"2014-05-04T09:31:05Z","abstract_excerpt":"Let $(X_1, \\xi_1), (X_2,\\xi_2),\\ldots$ be i.i.d.~copies of a pair $(X,\\xi)$ where $X$ is a random process with paths in the Skorokhod space $D[0,\\infty)$ and $\\xi$ is a positive random variable. Define $S_k := \\xi_1+\\ldots+\\xi_k$, $k \\in \\mathbb{N}_0$ and $Y(t) := \\sum_{k\\geq 0} X_{k+1}(t-S_k) 1_{\\{S_k \\leq t\\}}$, $t\\geq 0$. We call the process $(Y(t))_{t \\geq 0}$ random process with immigration at the epochs of a renewal process. We investigate weak convergence of the finite-dimensional distributions of $(Y(ut))_{u>0}$ as $t\\to\\infty$. Under the assumptions that the covariance function of $X$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.0671","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"}