{"paper":{"title":"Singular integrals of stable subordinator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Lihu Xu","submitted_at":"2018-01-24T01:32:56Z","abstract_excerpt":"It is well known that $\\int_{0}^{1} t^{-\\theta} d t<\\infty$ for $\\theta \\in (0,1)$ and $\\int_{0}^{1} t^{-\\theta} d t=\\infty$ for $\\theta \\in [1,\\infty)$. Since $t$ can be taken as an $\\alpha$-stable subordinator with $\\alpha=1$, it is natural to ask whether $\\int_{0}^{1} t^{-\\theta} d S_{t}$ has a similar property when $S_{t}$ is an $\\alpha$-stable subordinator with $\\alpha \\in (0,1)$. We show that $\\theta=\\frac 1\\alpha$ is the border line such that $\\int_{0}^{1} t^{-\\theta} d S_{t}$ is finite a.s. for $\\theta \\in (0, \\frac 1\\alpha)$ and blows up a.s. for $\\theta \\in [\\frac1\\alpha,\\infty)$. Wh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.07830","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"}