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

## TL;DR

This paper investigates the boundedness and integrability properties of reciprocals of probability generating functions, providing stronger counterexamples and systematic analysis, with implications for renewal theory and continuous-time processes.

## Contribution

It offers new, stronger counterexamples to a classical problem and systematically studies $L^p$-integrability of reciprocals under various conditions.

## Key findings

- Boundedness of reciprocals generally fails.
- Reciprocals exhibit certain $L^p$-integrability properties.
- Results extend to continuous-time settings.

## 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.