Resumen de On Erdős–de Bruijn–Kingman’s problem on regularity of reciprocals for exponential series

Alexander Gomilko, Yuri Tomilov

• Motivated by applications to renewal theory, Erdős, de Bruijn and Kingman posed a problem on boundedness of reciprocals (1−z)/(1−F(z)) in the unit disc for probability generating functions F(z). This problem was solved by Ibragimov in 1975 by constructing a counterexample. In this paper, we provide much stronger counterexamples showing that the problem does not allow for a positive answer even under rather restrictive additional assumptions. Moreover, we pursue a systematic study of Lp-integrabilty properties for the reciprocals. In particular, we show that while the boundedness of (1−z)/(1−F(z)) fails in general, the reciprocals do possess certain Lp-integrability properties under mild conditions on F. We also study the same circle of problems in the continuous-time setting.