{"id":340538,"date":"2016-12-22T09:15:09","date_gmt":"2016-12-22T17:15:09","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=340538"},"modified":"2018-10-16T21:23:15","modified_gmt":"2018-10-17T04:23:15","slug":"poisson-splitting-factors","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/poisson-splitting-factors\/","title":{"rendered":"Poisson Splitting by Factors"},"content":{"rendered":"<p>Given a homogeneous Poisson process on R<em><sup>d<\/sup><\/em> with intensity \u03bb, we prove that it is possible to partition the points into two sets, as a deterministic function of the process, and in an isometry-equivariant way, so that each set of points forms a homogeneous Poisson process, with any given pair of intensities summing to \u03bb. In particular, this answers a question of Ball [3], who proved that in <em>d<\/em> = 1, the Poisson points may be similarly partitioned (via a translation-equivariant function) so that one set forms a Poisson process of lower intensity, and asked whether the same was possible for all <em>d<\/em>. We do not know whether it is possible similarly to add points (again chosen as a deterministic function of a Poisson process) to obtain a Poisson process of higher intensity, but we prove that this is not possible under an additional finitariness condition<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Given a homogeneous Poisson process on Rd with intensity \u03bb, we prove that it is possible to partition the points into two sets, as a deterministic function of the process, and in an isometry-equivariant way, so that each set of points forms a homogeneous Poisson process, with any given pair of intensities summing to \u03bb. 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