By Aleksander Janicki

Offers new laptop equipment in approximation, simulation, and visualization for a number of alpha-stable stochastic procedures.

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Extra resources for Simulation and chaotic behavior of [alpha]-stable stochastic processes

Example text

2) and p E (0,0), then , and if p c [Q, 2), then 5. (a, 0, p) and a E (1,2], Ex = Covariation. then u Let (Xi. X 2 ) denote a jointly SaS random vector, where c ( 1 , 2]. Considering the Sati random variable Y = 0 1 X 1 0 2 X 2 for any real 25 A. JANICKI and A. WERON 0 1 , 0 2 we get Y 0, 0) with a = 0- (01, 02). -1• The covariation is designed to replace the covariance when a E (1, 2). In the case of a = 2 we have the following relation between these two expressions [ X1, X2]2 = Co v (X 1 , X2)• Asymptotic behavior of tail probabilities.

JANICKI and A. „ construct their , for j = 1,2, ... , n; statistical samples {Csi m ' m=-1 • notice that each 7 7 can be represented as a sum of j exponentially distributed independent random variables with mean parameter A = 1, so the method of computation of samples obvious; { rim) M for j = 1,2, ... ,n is • construct as a final result the corresponding sample for the sum defining X„. Unfortunately, as we shall in the next section, this algorithm is very costly as far as the time of calculations is concerned, even for small values of n.

The algorithm describing them involves only one deterministic function of two real variables. Thanks to some computer experiments we were able to provide a deeper quantitative insight into the structure of stable laws, varying with some parameters defining them. , Tapia and Thompson (1978)). Computer methods of constructing stochastic processes involve at least two kinds of discretization techniques: discretization of the time parameter and approximate representation of random variates with the aid of artificially produced finite time series data sets or statistical samples so we arc interested in statistical methods of data analysis such as constructions of empirical cumulative distribution functions or kernel probability density estimates, etc.

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