On the problem by Erdös-de Bruijn-Kingman 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 $F(z)$. 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 mild conditions on $F$. We also study the same circle of problems in the continuous-time setting.

38 pages, This is a version of the paper to appear in Revista Matemática Iberoamericana

References in corpus (2)

• Rational weak mixing in infinite measure spaces
• Conditions for rational weak mixing