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

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##### Citation

Ewerhart, C., & Serena, M. (2024). On the (Im-)Possibility of Representing Probability
Distributions as a Difference of I.I.D. Noise Terms.* Mathematics of Operations Research,*
forthcoming. doi:10.1287/moor.2023.0081.

Cite as: https://hdl.handle.net/21.11116/0000-000F-3D68-9

##### 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.