Characterization and analysis of generalized grey incomplete gamma noise

Abstract

The grey incomplete gamma distributions was established by one of the authors in a previous publication. In this article we use the Kondratiev characterization theorem to identify those via a suitable Laplace transform with holomorphic functions with suitable properties. We establish theorems for the integration and convergence of sequences of these distributions. As direct applications of these analytic tools we give the examples of Donsker's delta function, identify the time-derivative of the process as a suitable distribution and define the Gamma grey Ornstein–Uhlenbeck process.

Keywords:

• Non-Gaussian analysis
• gamma-grey noise
• fractional Ornstein-Uhlenbeck process
• characterization theorems
• Donsker delta function

Acknowledgments

The authors thank J. L. da Silva for his valuable discussions about the project. L.C. wants to thank the Centro de Ciências Matemáticas for the kind hospitality within his research stay in Funchal in spring 2023. We are grateful for the fruitful comments of L. Accardi which improved the study. We also would thank the reviewers whose suggestions improved the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Conflict of interest statement/ data accessibility statement

The authors declare that they have NO affiliations with or involvement in any organization or entity with any financial interest (such as honoraria; educational grants; participation in speakers' bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent-licensing arrangements), or non-financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript. There is no data used in the manuscript.