By Hideyuki Usui; Y Omura (eds.)

It is a selection of prolonged lecture notes of the tutorials given on the foreign college for area Simulations (ISSS)-7, March 2005, by way of the invited teachers who've been actively taken with machine simulation innovations in house plasma physics

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However, as it was mentioned previously, the usage of the stable distributions in financial applications is often restricted because of their infinite variance at a < 2. The compromise that retains flexibility of the Levy Probability Distributions 27 distribution yet yields finite variance is named truncated Levy flight. 9), fL (x) is the Levy distribution ‘ is the cutoff length, and C is the normalization constant. Sometimes the exponential cut-off is used at large distances [3] fTL (x) $ exp ( À ljxj), l > 0, jxj > ‘ (3:3:10) Since fTL (x) has finite variance, it converges to the normal distribution according to the central limit theorem.

Covariance between two variates provides a measure of their simultaneous change. Consider two variates, X and Y, that have the means mX and mY , respectively. Their covariance equals Cov(x, y)  sXY ¼ E[(x À mX )(y À mY )] ¼ E[xy] À mX mY (3:1:13) Obviously, covariance reduces to variance if X ¼ Y: sXX ¼ sX 2 . Positive covariance between two variates implies that these variates tend to change simultaneously in the same direction rather than in opposite directions. Conversely, negative covariance between two variates implies that when one variate grows, the second one tends to fall and vice versa.

However, special methods have been developed for their estimation [5]. It is said that the cumulative distribution function F(x) is in the domain of attraction of Hj (x). The tail behavior of the distribution F(x) defines the shape parameter. The Gumbel distribution corresponds to the case with j ¼ 0. Distributions with thin tails, such as normal, lognormal, and exponential distributions, have the Gumbel domain of attraction. The case with j > 0 is named the Frechet distribution. Domain of the Frechet attraction corresponds to distributions with fat tails, such as the Cauchy distribution and the Pareto distribution (see the next Section).

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