On Erdős–de Bruijn–Kingman’s problem on regularity of reciprocals for exponential series
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Alexander Gomilko [1] ; Yuri Tomilov [2]
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[1] Nicolaus Copernicus University
Nicolaus Copernicus University
Toruń, Polonia
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[2] Polish Academy of Sciences
Polish Academy of Sciences
Warszawa, Polonia
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- Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 37, Nº 3, 2021, págs. 1045-1081
- Idioma: inglés
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Resumen
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 Lp-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 Lp-integrability properties under mild conditions on F. We also study the same circle of problems in the continuous-time setting.
