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Discrete Analogs of Wigner Transforms and Weyl Transforms Read chapter Read first chapter

2019 | OriginalPaper | Chapter

Operator Transformation of Probability Densities

Author : Leon Cohen

Published in: Analysis of Pseudo-Differential Operators

Publisher: Springer International Publishing

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Abstract

We describe an operator relation that relates two arbitrary probability densities. The relation may be thought of as a generalization of the Edgeworth and Gram–Charlier series of probability theory. We apply the relation to a number of issues. We generalize to relate a probability distribution with itself but at different times and show that it can be used to obtain approximate solutions. We apply the scale operator to the case of the product of two independent random variables, and generalize the concept of cumulants for that case. An operator relation between the energy density of a signal and the energy density of the spectrum is obtained. In addition, we show that the spectral moments of a signal may be expressed in terms of the Bell polynomials.

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previous chapter Bismut’s Way of the Malliavin Calculus for Non-Markovian Semi-groups: An Introduction
next chapter The Time-Frequency Interference Terms of the Green’s Function for the Harmonic Oscillator
Appendix
Available only for authorised users
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Metadata
Title
Operator Transformation of Probability Densities
Author
Leon Cohen
Copyright Year
2019
Book
Analysis of Pseudo-Differential Operators

Print ISBN: 978-3-030-05167-9

Electronic ISBN: 978-3-030-05168-6

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-05168-6_8

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