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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.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 21, 2008
PubMed
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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.

Related Experiment Videos

  • 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.