JournalsrmiVol. 37, No. 3pp. 1045–1081

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

  • Alexander Gomilko

    Nicolaus Copernicus University, Torun, Poland and National Academy of Sciences of Ukraine, Kyiv, Ukraine
  • Yuri Tomilov

    Polish Academy of Sciences, Warsaw, Poland
On Erdős–de Bruijn–Kingman’s problem on regularity of reciprocals for exponential series cover
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Abstract

Motivated by applications to renewal theory, Erdős, de Bruijn and Kingman posed a problem on boundedness of reciprocals (1z)/(1F(z))(1 − z)/(1 − F(z)) in the unit disc for probability generating functions F(z)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 LpL^p-integrabilty properties for the reciprocals. In particular, we show that while the boundedness of (1z)/(1F(z))(1 − z)/(1 − F(z)) fails in general, the reciprocals do possess certain LpL^p-integrability properties under mild conditions on FF. We also study the same circle of problems in the continuous-time setting.

Cite this article

Alexander Gomilko, Yuri Tomilov, On Erdős–de Bruijn–Kingman’s problem on regularity of reciprocals for exponential series. Rev. Mat. Iberoam. 37 (2021), no. 3, pp. 1045–1081

DOI 10.4171/RMI/1220