Statistical Science

Can One See $\alpha$-Stable Variables and Processes?

Aleksander Janicki and Aleksander Weron

Full-text: Open access

Abstract

In this paper, we demonstrate some properties of $\alpha$-stable (stable) random variables and processes. It turns out that with the use of suitable statistical estimation techniques, computer simulation procedures and numerical discretization methods it is possible to construct approximations of stochastic integrals with stable measures as integrators. As a consequence we obtain an effective, general method giving approximate solutions for a wide class of stochastic differential equations involving such integrals. Application of computer graphics provides interesting quantitative and visual information on those features of stable variates which distinguish them from their commonly used Gaussian counterparts. It is possible to demonstrate evolution in time of densities with heavy tails of appropriate processes, to visualize the effect of jumps of trajectories, etc. We try to demonstrate that stable variates can be very useful in stochastic modeling of problems of different kinds, arising in science and engineering, which often provide better description of real life phenomena than their Gaussian counterparts.

Article information

Source
Statist. Sci., Volume 9, Number 1 (1994), 109-126.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.ss/1177010656

Digital Object Identifier
doi:10.1214/ss/1177010656

Mathematical Reviews number (MathSciNet)
MR1278680

Zentralblatt MATH identifier
0955.60508

JSTOR
links.jstor.org

Keywords
Stable distributions stable processes stochastic integrals and differential equations with stable integrators statistical estimation stochatic modeling computer simulation

Citation

Janicki, Aleksander; Weron, Aleksander. Can One See $\alpha$-Stable Variables and Processes?. Statist. Sci. 9 (1994), no. 1, 109--126. doi:10.1214/ss/1177010656. https://projecteuclid.org/euclid.ss/1177010656


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