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Mean and minimum of independent random variables

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Abstract

We show that any pair X, Y of independent non-compactly supported random variables on [0, ∞) satisfies

$$\mathop {\lim \inf}\limits_{m \to \infty} \mathbb{P}(\min (X,Y) > m\,\left| {X + Y > 2m} \right.) = 0.$$

.

We conjecture multi-variate and weighted generalizations of this result, and prove them under the additional assumption that the random variables are identically distributed.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Bar-Ilan University, Ramat-Gan, 5290002, Israel

    Naomi Dvora Feldheim

  2. Einstein Institute of Mathematics, The Hebrew University of Jerusalem Givat Ram, Jerusalem, 9190401, Israel

    Ohad Noy Feldheim

Authors
  1. Naomi Dvora Feldheim
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  2. Ohad Noy Feldheim
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Corresponding author

Correspondence to Naomi Dvora Feldheim.

Additional information

Research supported in part by the Institute of Mathematics and Its Applications funded by the NSF, and by an NSF postdoctoral fellowship at Stanford University.

Research supported in part by the Institute of Mathematics and Its Applications funded by the NSF and by a postdoctoral fellowship at Stanford University.

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Feldheim, N.D., Feldheim, O.N. Mean and minimum of independent random variables. Isr. J. Math. 244, 857–882 (2021). https://doi.org/10.1007/s11856-021-2195-0

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  • DOI: https://doi.org/10.1007/s11856-021-2195-0

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