DOI: https://doi.org/10.15588/1607-3274-2016-3-3

### RELATIONSHIP BETWEEN THE PARAMETERS OF SELF-SIMILARITY, STABILITY AND LONG-RANGE DEPENDENCY OF FRACTAL LEV MOTION

V. L Shergin

#### Abstract

The problem of searching relationships between the parameters of self-similarity, stability and long-range dependency of fractal Lev motion
is considered. It was proposed to use indexes, constructed by means a model of symmetric mixing of latent factors as a measure of the
relationship between increments of fractal Lev motion process. This approach make it possible to solve the problem of inability to use the
correlation method for estimating such increments caused by the absence of the required distribution moments. Dependence of the relationship index of neighboring increments on the indices of stability and self-similarity is obtained. This dependence has the form of an algebraic equation which has no an explicit solution generally, but can be easily solved numerically. A mathematical model that allows to construct a discrete analogue of the autocorrelation function for the fractal Lev motion is proposed. This model has the form of the system of algebraic equations. It is shown that all of the similar dependencies known for the particular cases of the fractal Lev motion, are special cases of the models obtained in the work. Proposed models allows us to determine any of the three parameters (self-similarity, stability and long-range dependency) on two other that will essentially simplify the modeling and studying stochastic processes having the form of fractal Levý motion.

#### Keywords

fractal Lev motion, self-similarity, long-range dependency, stable distributions, relationship indexes.

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#### References

Mandelbrot B. Fractional Brownian motions, fractional noises and applications / B. Mandelbrot, J.W. van Ness // SIAM Review. – 1968. – Vol. 10 (4) – P. 422–437. DOI:10.1137/1010093. 2. Ширяев А. Н. Основы стохастической финансовой математики. Том. 1. Факты. Модели / А. Н. Ширяев. – М. : ФАЗИС, 1998. – 512 с. 3. Мандельброт Б. Фрактальная геометрия природы / Б. Мандельброт. – М. : «Институт компьютерных исследований», 2002. – 656 c. 4. Золотарев В. М. Одномерные устойчивые распределения / В. М. Золотарев. – М. : Наука, 1983. –304 с. 5. Федер Е. Фракталы / Е.Федер. – М. : Мир, 1991. – 261 с. 6. Samorodnitsky G. Stable Non-Gaussian Random Processes, Chapter 7: «Self-similar processes» / G. Samorodnitsky, M. S. Taqqu. – Chapman & Hall, 1994. 7. Шергин В. Интерпретация показателя взаимосвязи многомерных устойчивых случайных величин с помощью факторной модели / В. Шергин // Восточно-Европейский журнал передовых технологий. – 2015. – 5 (4(77)) – С. 44–49. DOI:10.15587/1729-4061.2015.50442 8. Press S. J. Multivariate stable distributions / S. J. Press // Journal of Multivariate Analysis. – 1972. – Vol. 2, Issue 4. – P. 444–462. DOI: 10.1016/0047-259x(72)90038-3 9. Balakrishnan N.·Continuous bivariate distributions [Text] / N. Balakrishnan, C.-D. Lai. – Springer, 2009. – 684 p. DOI: 10.1007/b101765

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