{"paper":{"title":"Almost Sure Convergence of Solutions to Non-Homogeneous Stochastic Difference Equation","license":"","headline":"","cross_cats":["math.DS"],"primary_cat":"math.PR","authors_text":"Alexandra Rodkina, Gregory Berkolaiko","submitted_at":"2005-08-19T16:35:03Z","abstract_excerpt":"We consider a non-homogeneous nonlinear stochastic difference equation\n  X_{n+1} = X_n (1 + f(X_n)\\xi_{n+1}) + S_n, and its important special case\n  X_{n+1} = X_n (1 + \\xi_{n+1}) + S_n, both with initial value X_0, non-random decaying free coefficient S_n and independent random variables \\xi_n. We establish results on \\as convergence of solutions X_n to zero. The necessary conditions we find tie together certain moments of the noise \\xi_n and the rate of decay of S_n. To ascertain sharpness of our conditions we discuss some situations when X_n diverges. We also establish a result concerning th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0508371","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"}