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Design and Use of Multiplexed Chemostat Arrays
Published on: February 23, 2013
Continuous and discrete stable processes
W H Lee1, K I Hopcraft, E Jakeman
1School of Mathematical Sciences, Applied Mathematics Division, University of Nottingham, Nottingham, NG7 2RD, United Kingdom.
Summary
One-sided Lévy-stable probability densities and discrete-stable distributions form a unique pair. This connection enables the creation of novel continuous-stable stochastic processes.
Area of Science:
- Probability theory
- Stochastic processes
- Mathematical physics
Background:
- Lévy-stable distributions are generalizations of the normal distribution with heavier tails.
- Discrete-stable distributions model phenomena with discrete steps and heavy tails.
- The doubly stochastic Poisson transform is a mathematical tool connecting different probability distributions.
Purpose of the Study:
- To establish a novel connection between one-sided Lévy-stable probability densities and discrete-stable distributions.
- To leverage this connection for the development of new continuous-stable stochastic processes.
Main Methods:
- Utilizing the concept of the doubly stochastic Poisson transform.
- Analyzing the mathematical properties of one-sided Lévy-stable and discrete-stable distributions.
- Formulating a new class of continuous-stable stochastic processes based on the identified transform pair.
Main Results:
- Demonstrated that one-sided Lévy-stable probability densities and discrete-stable distributions constitute a doubly stochastic Poisson transform pair.
- Established a theoretical framework for a class of continuous-stable stochastic processes derived from this relationship.
Conclusions:
- The identified transform pair provides a powerful link between continuous and discrete probability models.
- This work opens new avenues for modeling complex systems using continuous-stable stochastic processes.
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