Sharp and simple bounds for the raw moments of the binomial and Poisson distributions

Thomas D. Ahle

Statistics & Probability Letters, 2022, vol. 182, issue C

Abstract: We prove the inequality E[(X/μ)k]≤(k/μlog(1+k/μ))k≤exp(k2/(2μ)) for sub-Poissonian random variables X, such as Binomially or Poisson distributed variables, with mean μ. The asymptotic behaviour E[(X/μ)k]=1+O(k2/μ) matches a lower bound of 1+Ω(k2/μ) for small k2/μ. This improves over previous uniform raw moment bounds by a factor exponential in k.

Keywords: Random variables; Moments; Asymptotics (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1016/j.spl.2021.109306

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