On the Continuity of Characteristic Functionals and Sparse Stochastic Modeling

J. Fageot, A. Amini, M. Unser

The Journal of Fourier Analysis and Applications, vol. 20, no. 6, pp. 1179-1211, December 2014.

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The characteristic functional is the infinite-dimensional generalization of the Fourier transform for measures on function spaces. It characterizes the statistical law of the associated stochastic process in the same way as a characteristic function specifies the probability distribution of its corresponding random variable. Our goal in this work is to lay the foundations of the innovation model, a (possibly) non-Gaussian probabilistic model for sparse signals. This is achieved by using the characteristic functional to specify sparse stochastic processes that are defined as linear transformations of general continuous-domain white Lévy noises (also called innovation processes). We prove the existence of a broad class of sparse processes by using the Minlos-Bochner theorem. This requires a careful study of the regularity properties, especially the L^p-boundedness, of the characteristic functional of the innovations. We are especially interested in the functionals that are only defined for p < 1 since they appear to be associated with the sparser kind of processes. Finally, we apply our main theorem of existence to two specific subclasses of processes with specific invariance properties.

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AUTHOR="Fageot, J. and Amini, A. and Unser, M.",
TITLE="On the Continuity of Characteristic Functionals and Sparse
	Stochastic Modeling",
JOURNAL="The Journal of {F}ourier Analysis and Applications",
YEAR="2014",
volume="20",
number="6",
pages="1179--1211",
month="December",
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DOI: 10.1007/s00041-014-9351-4