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2022 Geometric sums, size biasing and zero biasing
Qingwei Liu, Aihua Xia
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Electron. Commun. Probab. 27: 1-13 (2022). DOI: 10.1214/22-ECP462

Abstract

The geometric sum plays a significant role in risk theory and reliability theory [Kalashnikov (1997)] and a prototypical example of the geometric sum is Rényi’s theorem [Rényi (1956)] saying a sequence of suitably parameterised geometric sums converges to the exponential distribution. There is extensive study of the accuracy of exponential distribution approximation to the geometric sum [Sugakova (1995), Kalashnikov (1997), Peköz & Röllin (2011)] but there is little study on its natural counterpart of gamma distribution approximation to negative binomial sums. In this note, we show that a nonnegative random variable follows a gamma distribution if and only if its size biasing equals its zero biasing. We combine this characterisation with Stein’s method to establish simple bounds for gamma distribution approximation to the sum of nonnegative independent random variables, a class of compound Poisson distributions and the negative binomial sum of random variables.

Acknowledgments

We thank Nathan Ross for suggesting the direct coupling proof in Remark 4.5 and in the context of infinitely divisible distributions on R+, Theorem 2.1 is indirectly confirmed in the literature. We also thank a referee for bringing to our attention that the zero-bias transformation dates back at least to [Lukacs (1970), Theorem 12.2.5 (a)].

Citation

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Qingwei Liu. Aihua Xia. "Geometric sums, size biasing and zero biasing." Electron. Commun. Probab. 27 1 - 13, 2022. https://doi.org/10.1214/22-ECP462

Information

Received: 16 September 2021; Accepted: 13 March 2022; Published: 2022
First available in Project Euclid: 21 March 2022

Digital Object Identifier: 10.1214/22-ECP462

Subjects:
Primary: 60E15 , 60F05

Keywords: gamma distribution , geometric sum , size-biasing , Stein’s method , zero-biasing

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