On the (Im-)Possibility of Representing Probability Distributions as a Difference of I.I.D. Noise Terms

Christian Ewerhart () and Marco Serena ()
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Christian Ewerhart: Department of Economics, University of Zurich, 8001 Zurich, Switzerland
Marco Serena: School of Business, Economics and Law, CUNEF Universidad, 28040 Madrid, Spain

Mathematics of Operations Research, 2025, vol. 50, issue 1, 390-409

Abstract: A random variable is difference-form decomposable ( DFD ) if it may be written as the difference of two i.i.d. random terms. We show that densities of such variables exhibit a remarkable degree of structure. Specifically, a DFD density can be neither approximately uniform, nor quasiconvex, nor strictly concave. On the other hand, a DFD density need, in general, be neither unimodal nor logconcave. Regarding smoothness, we show that a compactly supported DFD density cannot be analytic and will often exhibit a kink even if its components are smooth. The analysis highlights the risks for model consistency resulting from the strategy widely adopted in the economics literature of imposing assumptions directly on a difference of noise terms rather than on its components.

Keywords: Primary: 60E05; Secondary: 91B26; differences of random variables; density functions; characteristic function; uniform distribution (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:50:y:2025:i:1:p:390-409

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