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Motivated by applications to renewal theory, Erdős, de Bruijn and Kingman posed a problem on boundedness of reciprocals in the unit disc for probability generating functions . 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 -integrabilty properties for the reciprocals. In particular, we show that while the boundedness of fails in general, the reciprocals do possess certain -integrability properties under mild conditions on . We also study the same circle of problems in the continuous-time setting.
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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. 1045–1081DOI 10.4171/RMI/1220