Chebyshev Inequalities for Products of Random Variables

Napat Rujeerapaiboon (), Daniel Kuhn () and Wolfram Wiesemann ()
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Napat Rujeerapaiboon: École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
Daniel Kuhn: École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
Wolfram Wiesemann: Imperial College Business School, Imperial College London, London SW7 2AZ, United Kingdom

Mathematics of Operations Research, 2018, vol. 43, issue 3, 887-918

Abstract: We derive sharp probability bounds on the tails of a product of symmetric nonnegative 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 available, 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 derive Chebyshev bounds for sums, maxima, and minima of nonnegative random variables.

Keywords: Chebyshev inequality; probability bounds; distributionally robust optimization; convex optimization (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (4)

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