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February, 1994 Can One See $\alpha$-Stable Variables and Processes?
Aleksander Janicki, Aleksander Weron
Statist. Sci. 9(1): 109-126 (February, 1994). DOI: 10.1214/ss/1177010656


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.


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Aleksander Janicki. Aleksander Weron. "Can One See $\alpha$-Stable Variables and Processes?." Statist. Sci. 9 (1) 109 - 126, February, 1994.


Published: February, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0955.60508
MathSciNet: MR1278680
Digital Object Identifier: 10.1214/ss/1177010656

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.9 • No. 1 • February, 1994
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