On the problem by Erdös-de Bruijn-Kingman on regularity of reciprocals for exponential series

Abstract

Motivated by applications to renewal theory, Erdös, de Bruijn and Kingman posed in 50th-70th a problem on boundedness of reciprocals (1 − z)/(1 − F (z)) for probability generating functions F (z). It 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 L p -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 L p -integrability properties under rather mild conditions on F . We also study the same circle of problems in the continuous time setting.