Gaps in samples of geometric random variables

In this note we continue the study of gaps in samples of geometric random variables originated in Hitczenko and Knopfmacher [Gap-free compositions and gap-free samples of geometric random variables. Discrete Math.   294 (2005) 225–239] and continued in Louchard and Prodinger [The number of gaps in sequences of geometrically distributed random variables, Preprint available at 〈〈http://www.ulb.ac.be/di/mcs/louchard/〉〉 (number 81 on the list) or at 〈〈http://math.sun.ac.za/∼∼ prodinger/pdffiles/gapsAPRIL27.pdf.〉〉] In particular, since the notion of a gap differs in these two papers, we derive some of the results obtained in Louchard and Prodinger [The number of gaps in sequences of geometrically distributed random variables, Preprint available at 〈〈http://www.ulb.ac.be/di/mcs/louchard/〉〉 (number 81 on the list) or at 〈〈http://math.sun.ac.za/∼∼prodinger/pdffiles/gapsAPRIL27.pdf.〉〉] for gaps as defined in Hitczenko and Knopfmacher [Gap-free compositions and gap-free samples of geometric random variables. Discrete Math. 294 (2005) 225–239].