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arXiv · 1701.04357

On the problem by Erd\"os-de Bruijn-Kingman on regularity of reciprocals for exponential series

Alexander Gomilko·Yuri Tomilov
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Abstract

Motivated by applications to renewal theory, Erd\H{o}s, 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.

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Alexander Gomilko, Yuri Tomilov. 2017-01-16. On the problem by Erd\"os-de Bruijn-Kingman on regularity of reciprocals for exponential series. https://arxiv.org/abs/1701.04357

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