{"paper":{"title":"On the Integral of Fractional Poisson Processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Enzo Orsingher, Federico Polito","submitted_at":"2013-03-26T22:35:53Z","abstract_excerpt":"In this paper we consider the Riemann--Liouville fractional integral $\\mathcal{N}^{\\alpha,\\nu}(t)= \\frac{1}{\\Gamma(\\alpha)} \\int_0^t (t-s)^{\\alpha-1}N^\\nu(s) \\, \\mathrm ds $, where $N^\\nu(t)$, $t \\ge 0$, is a fractional Poisson process of order $\\nu \\in (0,1]$, and $\\alpha > 0$. We give the explicit bivariate distribution $\\Pr \\{N^\\nu(s)=k, N^\\nu(t)=r \\}$, for $t \\ge s$, $r \\ge k$, the mean $\\mathbb{E}\\, \\mathcal{N}^{\\alpha,\\nu}(t)$ and the variance $\\mathbb{V}\\text{ar}\\, \\mathcal{N}^{\\alpha,\\nu}(t)$. We study the process $\\mathcal{N}^{\\alpha,1}(t)$ for which we are able to produce explicit re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.6687","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"}