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Generating M-Indeterminate Probability Densities by Way of Quantum Mechanics

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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.

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Acknowledgements

R. Sala Mayato acknowledges funding by the Spanish MINECO and FEDER, grant FIS2017-82855-P (MINE CO/ FEDER, UE), and by M. Payne.

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Authors and Affiliations

  1. Departamento de Física and IUdEA, Universidad de La Laguna, 38203, La Laguna, Tenerife, Spain

    Rafael Sala Mayato

  2. Departments of Bioengineering, and Electrical & Computer Engineering, University of Pittsburgh, Pittsburgh, PA, 15261, USA

    Patrick Loughlin

  3. Department of Physics, Hunter College of the City University of New York, 695 Park Ave., New York, NY, 10065, USA

    Leon Cohen

Authors
  1. Rafael Sala Mayato
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  2. Patrick Loughlin
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  3. Leon Cohen
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Correspondence to Leon Cohen.

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Mayato, R.S., Loughlin, P. & Cohen, L. Generating M-Indeterminate Probability Densities by Way of Quantum Mechanics. J Theor Probab 35, 1537–1555 (2022). https://doi.org/10.1007/s10959-021-01121-5

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  • DOI: https://doi.org/10.1007/s10959-021-01121-5

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