On random sets connected to the partial records of Poisson point process

Random intervals are constructed from partial records in a Poisson point process in] 0,infinity[x] 0,infinity[. These are used to cover partially [ 0,infinity[; the purpose of this work is to study the random set R that is left uncovered. We show that R enjoys the regenerative property and identify its distribution in terms of the characteristics of the Poisson point process. As an application we show that R is almost surely a fractal set and we calculate its dimension.