Generating M-Indeterminate Probability Densities by Way of Quantum Mechanics

Rafael Sala Mayato, Patrick Loughlin and Leon Cohen ()
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Rafael Sala Mayato: Universidad de La Laguna
Patrick Loughlin: University of Pittsburgh
Leon Cohen: Hunter College of the City University of New York

Journal of Theoretical Probability, 2022, vol. 35, issue 3, 1537-1555

Abstract: Abstract Probability densities that are not uniquely determined by their moments are said to be “moment-indeterminate,” or “M-indeterminate.” Determining whether or not a density is M-indeterminate, or how to generate an M-indeterminate density, is a challenging problem with a long history. Quantum mechanics is inherently probabilistic, yet the way in which probability densities are obtained is dramatically different in comparison with standard probability theory, involving complex wave functions and operators, among other aspects. Nevertheless, the end results are standard probabilistic quantities, such as expectation values, moments and probability density functions. We show that the quantum mechanics procedure to obtain densities leads to a simple method to generate an infinite number of M-indeterminate densities. Different self-adjoint operators can lead to new classes of M-indeterminate densities. Depending on the operator, the method can produce densities that are of the Stieltjes class or new formulations that are not of the Stieltjes class. As such, the method complements and extends existing approaches and opens up new avenues for further development. The method applies to continuous and discrete probability densities. A number of examples are given.

Keywords: M-indeterminate probability densities; Moments; Quantum mechanics; Stieltjes class; 44A60; 47N30; 60E05 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-021-01121-5

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