A sharp estimate for probability distributions

Stefan Steinerberger

Statistics & Probability Letters, 2019, vol. 155, issue C, -

Abstract: We consider absolutely continuous probability distributions f(x)dx on R≥0. A result of Feldheim & Feldheim shows, among other things, that if the distribution is not compactly supported, then, using X,Y to denote two independent random variables drawn from f(x)dx, there exists z>0 such that most events in X+Y≥2z are comprised of a ‘small’ term satisfying min(X,Y)≤z and a ‘large’ term satisfying max(X,Y)≥z (as opposed to two ‘large’ terms that are both larger than z). More formally, they showed lim supz→∞Pmin(X,Y)≤z|X+Y≥2z=1.The result fails if the distribution is compactly supported. We prove supz>0Pmin(X,Y)≤z|X+Y≥2z≥124+8log2(med(X)‖f‖L∞),where medX denotes the median. Interestingly, the logarithm is necessary and the result is sharp up to constants; we also discuss some open problems.

Keywords: Distribution-free estimate; Comparison inequalities; Geometry of probability distributions (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1016/j.spl.2019.108584

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