Estimate of the concentration function for a class of additive functions

Khripunova, M. B.; Yudin, A. A.

doi:10.1134/s0001434607090301

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Regularity Of Reciprocals In Terms Of the L P -Scale1

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2017

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“…for an absolute constant C > 0, see e.g. [4], [9], [10], [24], [29]. The estimates are motivated by applications in probability theory and number theory, and they seem to be weaker than the estimate (5.1) provided by Theorem 5.1.…”

Section: Regularity Of Reciprocals In Terms Of the L P -Scalementioning

confidence: 99%

On the problem by Erdös-de Bruijn-Kingman on regularity of reciprocals for exponential series

Gomilko,

Tomilov

2017

Preprint

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Motivated by applications to renewal theory, Erdös, de Bruijn and Kingman posed in 50th-70th a problem on boundedness of reciprocals (1 − z)/(1 − F (z)) 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 rather mild conditions on F . We also study the same circle of problems in the continuous time setting.

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Section: Regularity Of Reciprocals In Terms Of the L P -Scalementioning

confidence: 99%