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Chebychev inequalities for products of random variables

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Title: Chebychev inequalities for products of random variables
Authors: Rujeerapaiboon, N
Kuhn, D
Wiesemann, W
Item Type: Journal Article
Abstract: We derive sharp probability bounds on the tails of a product of symmetric non-negative random variables using only information about their first two moments. If the covariance matrix of the random variables is known exactly, these bounds can be computed numerically using semidefinite programming. If only an upper bound on the covariance matrix is avail- able, the probability bounds on the right tails can be evaluated analytically. The bounds under precise and imprecise covariance information coincide for all left tails as well as for all right tails corresponding to quantiles that are either sufficiently small or sufficiently large. We also prove that all left probability bounds reduce to the trivial bound 1 if the number of random variables in the product exceeds an explicit threshold. Thus, in the worst case, the weak-sense geometric random walk defined through the running product of the random variables is absorbed at 0 with certainty as soon as time exceeds the given threshold. The techniques devised for constructing Chebyshev bounds for products can also be used to de- rive Chebyshev bounds for sums, maxima and minima of non-negative random variables.
Issue Date: 1-Aug-2018
Date of Acceptance: 17-Jun-2017
URI: http://hdl.handle.net/10044/1/49438
DOI: https://dx.doi.org/10.1287/moor.2017.0888
ISSN: 1526-5471
Publisher: INFORMS (Institute for Operations Research and Management Sciences)
Start Page: 693
End Page: 1050
Journal / Book Title: Mathematics of Operations Research
Volume: 43
Issue: 3
Copyright Statement: © 2018, INFORMS
Sponsor/Funder: Engineering & Physical Science Research Council (E
Funder's Grant Number: EP/M028240/1
Keywords: Science & Technology
Technology
Physical Sciences
Operations Research & Management Science
Mathematics, Applied
Mathematics
Chebyshev inequality
probability bounds
distributionally robust optimization
convex optimization
DISTRIBUTIONALLY ROBUST OPTIMIZATION
UNCERTAINTY QUANTIFICATION
CONVEX-OPTIMIZATION
0102 Applied Mathematics
0103 Numerical And Computational Mathematics
0802 Computation Theory And Mathematics
Operations Research
Publication Status: Published
Online Publication Date: 2018-02-16
Appears in Collections:Imperial College Business School