# Characterization and Analysis of Generalized Grey Incomplete Gamma Noise

## TL;DR

This paper characterizes and analyzes generalized grey incomplete gamma noise using Laplace transforms, establishing convergence theorems and applying these to stochastic processes like the Ornstein-Uhlenbeck process.

## Contribution

It introduces a novel analytic framework for generalized grey incomplete gamma noise and applies it to stochastic process analysis.

## Key findings

- Established convergence theorems for these distributions
- Identified Donsker's delta and local time within this framework
- Defined the Gamma grey Ornstein-Uhlenbeck process

## 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, the local time, identify the time-derivative of the process as a suitable distribution and define the Gamma grey Ornstein-Uhlenbeck process.