A Distributional Approach to Generalized Stochastic Processes on Locally Compact Abelian Groups

Abstract: This paper is dedicated to Paul Butzer on the occasion of his 85th birthday. His work and example have strongly influenced not only the first author, but also generations of mathematicians working in approximation theory and Fourier analysis. He has shown younger colleagues the importance of remaining open to applied areas, avoiding an overly narrow scope, and exploring different ways of understanding mathematical facts. A recurring theme in his work is the logical equivalence of fundamental statements in analysis. It may be less widely known that, besides his central role in approximation theory, Paul Butzer has also made significant contributions to probability theory. We hope that he will enjoy this note, which shows that a purely functional-analytic treatment of generalized stochastic processes is possible. The approach is based on the Segal algebra S0(G) and avoids several technical difficulties associated with the customary framework of vector-valued integration and topological vector spaces.

• The paper introduces a distributional framework using the Segal algebra S0(G) to define generalized stochastic processes on LCA groups, addressing challenges in classical integration methods.
• It establishes key equivalences between stationarity, V-boundedness, and covariance properties through harmonic analysis and Fourier invariance of S0(G).
• The results extend classical theories, offering practical insights for applications in operator sampling, time-frequency analysis, and random signal processing.

A Distributional Framework for Generalized Stochastic Processes on Locally Compact Abelian Groups

Abstract and Motivation

The paper "A Distributional Approach to Generalized Stochastic Processes on Locally Compact Abelian Groups" (2606.31316) rigorously formulates generalized stochastic processes (GSPs) in the context of locally compact Abelian groups (LCA groups) using the Segal algebra ${S_{0}(G)}$ as a space of test functions. This approach circumvents technical complications encountered with classical vector-valued integration or topological vector space theory and leverages the algebraic and Fourier invariance properties of ${S_{0}(G)}$. The authors build a formalism that unifies stochastic process theory, harmonic analysis, and distribution theory, offering clarified definitions and structural results for stationary, harmonizable, and V-bounded processes.

Segal Algebra ${S_{0}(G)}$: Rationale and Properties

Central to the paper is the Segal algebra ${S_{0}(G)}$, a Banach space subalgebra of the Fourier algebra $A(G)$, characterized by translation and modulation invariance. Its construction hinges on compactly supported, continuous, and absolutely summable functions and can equivalently be described using the short-time Fourier transform (STFT) with the Gaussian function as window. Key features include:

• Fourier Invariance: ${S_{0}(G)}$ is mapped onto $S_{0}(\hat{G})$ under the Fourier transform, allowing for symmetric spectral and spatial representations.
• Minimality and Density: It is the smallest Banach space invariant under translations and modulations, containing $L^1(G)$ functions with compactly supported transforms. The Schwartz(-Bruhat) space is embedded densely within ${S_{0}(G)}$.
• Tensor Product Structure: The functorial property $$S_0(G_1 \times G_2) = S_0(G_1) \hat{\otimes} S_0(G_2)$$ is fundamental for kernel theorems and for defining GSPs with multivariate domains.

These properties are deployed to facilitate operator extension, define distributions, and ensure the applicability of harmonic analysis tools.

Generalized Stochastic Processes: Definitions and Structural Results

A generalized stochastic process is defined as a bounded linear mapping ${S_{0}(G)}$0, where ${S_{0}(G)}$1 is a Hilbert space of mean-zero, square-integrable random variables. Classical concepts are reinterpreted:

• Stationarity: ${S_{0}(G)}$2 is stationary if its inner products are invariant under translation in ${S_{0}(G)}$3.
• Frequency Stationarity: Invariance under modulation by characters on ${S_{0}(G)}$4.
• Orthogonally Scattered: Support disjointness of test functions implies orthogonality in ${S_{0}(G)}$5.

The autocovariance distribution ${S_{0}(G)}$6 is constructed as a bilinear form on the projective tensor product ${S_{0}(G)}$7, and extended to the distributional dual space when appropriate.

Covariance Structure and Characterizations

The equivalence between the properties of the GSP ${S_{0}(G)}$8 and its covariance ${S_{0}(G)}$9 is proven in detail. Fundamental equivalences include:

• Stationarity ${S_{0}(G)}$0 Diagonal Invariance of Covariance: ${S_{0}(G)}$1 is stationary if and only if ${S_{0}(G)}$2 is invariant under simultaneous translation in both variables.
• Boundedness ${S_{0}(G)}$3 Covariance Extends to a Bimeasure: The covariance extends uniquely to a bimeasure iff ${S_{0}(G)}$4 is bounded.
• Orthogonally Scattered ${S_{0}(G)}$5 Diagonal Support of Covariance: The covariance is supported on the diagonal ${S_{0}(G)}$6 iff ${S_{0}(G)}$7 is orthogonally scattered; in this case, it admits a representation via a positive, translation-bounded measure.

Covariance structures are further related to the spectral measure via Fourier transforms, demonstrating the symmetry between time and frequency domains and enabling spectral representation theorems.

Harmonizable and V-Bounded Processes

The concepts of harmonizability and V-boundedness generalize classical stationarity. A GSP is harmonizable if its covariance is realized in the Fourier-Stieltjes algebra ${S_{0}(G)}$8, i.e., is the Fourier transform of a bounded measure. The paper proves:

• Harmonizability ${S_{0}(G)}$9 V-Boundedness: All harmonizable GSPs are V-bounded.
• Equivalence for Stationary GSPs: For stationary GSPs, harmonizability and V-boundedness are equivalent, linking process regularity to covariance algebraic structure.

A constructive result shows any V-bounded GSP can be approximated by harmonizable ones, with convergence of their autocovariance functions uniform on compact sets.

Relations to Classical Stochastic Processes and Vector Measures

The distributional framework aligns closely with classical stochastic processes when the covariance distribution can be represented by continuous, bounded functions. Key results:

• Bijective Correspondence: Continuous and bounded stochastic processes on ${S_{0}(G)}$0 correspond uniquely to GSPs with continuous bounded covariance via extension to measure spaces.
• Extension to Vector Measures: Under boundedness or V-boundedness, GSPs and vector measures are equivalent, establishing compatibility with Niemi's approach.
• Spectral Representation: V-bounded processes admit representation as Fourier transforms of stochastic measures, incorporating the classical spectral theorem.

Dilation Theory

The paper concludes with an extension of the dilation theorem, stating that every V-bounded GSP is the projection of a stationary GSP—thus any continuous V-bounded stochastic process admits a stationary dilation. This result is obtained via duality and the functional properties of ${S_{0}(G)}$1 and its Fourier transform.

Implications and Future Directions

Practically, this distributional approach streamlines the analysis and classification of stochastic processes in abstract group settings, with clear extension to time-frequency analysis, operator sampling, and random signal analysis. Theoretically, it clarifies the interplay between harmonic analysis, stochastic process theory, and distribution theory, especially in non-Euclidean domains.

Future research may pursue:

• Applications to Stochastic Operator Sampling: Leveraging modulation spaces and the Segal algebra for identification of stochastic operators, as indicated by related work on stochastic modulation spaces.
• Extension to Non-Abelian Groups: Investigating the feasibility and limitations of analogous frameworks in non-commutative settings.
• Algorithmic Realizations: Implementation strategies for stochastic process simulation and estimation in numerical harmonic analysis, exploiting the Banach space structure and Fourier invariance.

Conclusion

This paper provides a rigorous, technically coherent foundation for generalized stochastic processes on LCA groups via the Segal algebra ${S_{0}(G)}$2. It demonstrates that key stochastic properties manifest as algebraic or distributional characteristics, simplifies many classical proofs, and facilitates a unifying perspective on stochastic analysis, harmonic analysis, and distribution theory. The results have both broad theoretical relevance and practical applicability in abstract signal processing and random operator theory.