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The paper treats time-frequency analysis of scalar-valued zero mean Gaussian stochastic processes on ℝd. We prove that if the covariance function belongs to the Feichtinger algebra S0(ℝ2d) then: (i) the Wigner distribution and the ambiguity function of the process exist as finite variance stochastic Riemann integrals, each of which defines a stochastic process on ℝ2d, (ii) these stochastic processes on ℝ2d are Fourier transform pairs in a certain sense, and (iii) Cohen's class, ie convolution of the Wigner process by a deterministic function Φ∈C(ℝ2d), gives a finite variance process, and if Φ∈S0(ℝ2d) then W∗Φ can be expressed multiplicatively in the Fourier domain

journal of function spaces and applications/journal of function spaces and applications

The random Wigner distribution of Gaussian stochastic processes with covariance in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>ℝ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>

The random Wigner distribution of Gaussian stochastic processes with covariance inS0(ℝ2d)