### References & Citations

# Mathematics > Probability

# Title: Geometric sums, size biasing and zero biasing

(Submitted on 6 Jun 2021 (v1), last revised 16 Oct 2021 (this version, v2))

Abstract: The geometric sum plays a significant role in risk theory and reliability theory \cite{Kala97} and a prototypical example of the geometric sum is R\'enyi's theorem~\cite{Renyi56} 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 \cite{Sugakova95,Kala97,PekozRollin11} 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.

## Submission history

From: Qingwei Liu [view email]**[v1]**Sun, 6 Jun 2021 02:59:17 GMT (20kb)

**[v2]**Sat, 16 Oct 2021 04:31:13 GMT (14kb)

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