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

Abstract

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 $L^p$-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 mild conditions on $F$. We also study the same circle of problems in the continuous-time setting.