On the Continuity of Characteristic Functionals and Sparse Stochastic Modeling

Abstract

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.

Interpretation of continuous-time autoregressive processes as random exponential splines

“…Note that can be arbitrarily small, hence this condition is satisfied for any probability law used in practice. If a Lévy exponent f of an infinitely divisible random variable has a finite-moment of order > 0, then it is called a Lévy-Schwartz exponent [13,Section 3].…”

“…Note the strong connection between (3) and (4), where the sum over n is replaced by an integral over t. When f (ω) is a Lévy-Schwartz exponent, it is known that the characteristic functional in (4) defines a generalized random process in S (R) [13,Section 3].…”

“…Hence, f n (ω) = λ n ( P n (ω) − 1) is the Lévy exponent of an impulsive noise w n according to (13).…”

“…• From the relation f n (ω) −→ n→∞ f (ω), one can deduce, using the techniques of [13], that P wn (ψ) −→ n→∞ P w (ψ) for every ψ ∈ R(R), the space of rapidly decaying function (see Section II-C).…”

Sampling and (sparse) stochastic processes: A tale of splines and innovation

“…While the family includes the white Gaussian noises of the traditional theory of stochastic processes, it is considerably richer, the great majority of its members being sparse [24]. An important point is that (3) only holds in the sense of distributions since the innovation w ∈ S (R) is too rough to have a classical pointwise interpretation [25]. If L is splineadmissible, then it is generally possible to invert this equation, which yields the formal solution…”