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Skellam Type Processes of Order k and Beyond
Neha Gupta1, Arun Kumar1, Nikolai Leonenko2
1Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, Punjab 140001, India.
Entropy (Basel, Switzerland)
|December 8, 2020
Summary
We introduce the Skellam process of order k and its time-changed variants, including space-fractional and tempered versions. These novel processes generalize existing models and offer new insights into stochastic processes.
Area of Science:
- Probability theory
- Stochastic processes
- Mathematical analysis
Background:
- The Skellam distribution models the difference between two independent Poisson random variables.
- Generalizations of the Skellam process are essential for modeling complex phenomena.
Purpose of the Study:
- Introduce the Skellam process of order k and its running average.
- Investigate time-changed Skellam processes, including space-fractional and tempered versions.
- Derive key properties of these novel stochastic processes.
Main Methods:
- Development of the Skellam process of order k and its running average.
- Application of time changes using independent stable and tempered stable subordinators.
- Derivation of marginal probabilities, Lévy measures, and governing equations.
Main Results:
- Introduction of the Skellam process of order k and its running average.
- Characterization of time-changed Skellam processes, including space-fractional and tempered variants.
- Derivation of marginal distributions, Lévy measures, and difference-differential equations.
Conclusions:
- The introduced processes generalize the classical Skellam process and the running average of the Poisson process.
- The study provides a comprehensive analysis of the properties of these novel stochastic models.
- This work offers a foundation for further research in fractional and tempered stochastic processes.
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