**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**31103

##### Applications of Stable Distributions in Time Series Analysis, Computer Sciences and Financial Markets

**Authors:**
Mohammad Ali Baradaran Ghahfarokhi,
Parvin Baradaran Ghahfarokhi

**Abstract:**

In this paper, first we introduce the stable distribution, stable process and theirs characteristics. The a -stable distribution family has received great interest in the last decade due to its success in modeling data, which are too impulsive to be accommodated by the Gaussian distribution. In the second part, we propose major applications of alpha stable distribution in telecommunication, computer science such as network delays and signal processing and financial markets. At the end, we focus on using stable distribution to estimate measure of risk in stock markets and show simulated data with statistical softwares.

**Keywords:**
SAS,
VaR,
stable distribution,
infinite variance,
heavy tail networks

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1077597

**References:**

[1] Achim, A., A. Bezerianos, and P. Tsakalides (2002). SAR image denoising: a multiscale robust statistical approach. IEEE Proc. 14-th Intl. Conf. on Digital Signal Processing (DSP 2002), Santorini, Greece II, 1235{1238.

[2] Achim, A., P. Tsakalides, and A. Bezerianos (2003). SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modeling. IEEE Transactions on Geoscience and Remote Sensing 41, 1773{1784.

[3] _ B. V. Gnedenko and A. N. Kolmogorov (1954). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley.

[4] Cambanis, S., R. Keener, and G. Simons (1983). On ®- symmetric multivariate distributions. J. Multivar. Anal. 13, 213{233

[5] Cambanis, S. and M. Maejima (1989). Two classes of selfsimilar stable processes with stationary increments. Stoch. Proc. Appl. 32, 305{329.

[6] Cambanis, S., M. Maejima, and G. Samorodnitsky (1990). Characterization of linear and harmonizable fractional stable motions. Preprint.

[7] Cambanis, S. and E. Masry (1991). Wavelet approximation of deterministic and random signals: Convergence properties and rates. Technical Report 352, Center for Stochastic Processes, Univ. of North Carolina.

[8] Fama, E. (1965). The behavior of stock market prices. Journal of Business 38, 34{105.

[9] Fama, E. (1971). Risk, return and equilibrium. The Journal of Political Economy 79, 30{55.

[10] Fama, E. and R. Roll (1968). Some properties of symmetric stable distributions. JASA 63, 817{83.

[11] Fama, E. and R. Roll (1971). Parameter estimates for symmetric stable distributions. Journal of the American Statistical Association 66, 331{338.

[12] GNU Scientific Library - Reference Manual Edition 1.6, for GSL Version 1.6, 27 December 2004

[13] Levy, J. and M. Taqqu (1991). A characterization of the asymptotic behavior of stationary stable processes. In S. Cambanis, G. Samorodnitsky, and

[14] M. Taqqu (Eds.), Stable Processes and Related Topics, Volume 25 of Progress in Probability, Boston, pp. 181{198. Birkh├äauser.

[15] Levy., J. and M. Taqqu (2005, November). The asymptotic codi®erence and covariation of log-fractional stable noise. Preprint, BundesBank November 2005 Conference.

[16] L┬Âevy, P. (1924). Th┬Âeorie des erreurs la loi de Gauss et les lois exceptionelles. Bulletin Soc. Math. France 52, 49{85.

[17] L┬Âevy, P. (1925). Calcul des Probabilit┬Âes. Paris: Gauthier- Villars.

[18] L┬Âevy, P. (1954). Th┬Âeorie de l'addition des variables al┬Âeatoires. Paris: Gauthier- Villars. Originally appeared in 1937

[19] Mandelbrot, B. (1961). Stable Paretian random functions and the multiplicative variation of income. Econometria 29, 517{543.

[20] Mandelbrot, B. (1962). Sur certain prix sp┬Âeculatifs: faits empiriques et mod┬Âele bas┬Âe sur les processes stables additifs de Paul L┬Âevy. Comptes Rendus. 254, 3968{3970.

[21] Mandelbrot, B. (1963a). New methods in statistical economics. Journal of Political Econ. 71, 421{440.

[22] Mandelbrot, B. (1963b). The variation of certain speculative prices. Journal of Business 26, 394{419.

[23] Mandelbrot, B. (1965a). Self-similar error clusters in communications systems and the concept of conditional systems and the concept of conditional stationarity. I.E.E.E. Trans of Communications Technology COM-13, 71{90.

[24] Mandelbrot, B. (1965b). Une classe de processus stochastiques homothetiques a soi; application a loi climatologique de H.E. Hurst. Comptes Rendus Acad. Sci. Paris 240, 3274{3277.

[25] Mandelbrot, B. (1967). The variation of some other speculative prices. Journal of Business 40, 393{413.

[26] Mandelbrot, B. (1969). Long-run linearity, locally Gaussian processes, H-spectra and in┬»nite variances. International Econom. Rev. 10, 82{113.

[27] Mandelbrot, B. (1971). A fast fractional Gaussian noise generator. Water Resour. Res. 7, 543{553.

[28] Mandelbrot, B. (1972a). Broken line process derived as an approximation to fractional noise. Water Resour. Res. 8, 1354{1356.

[29] Mandelbrot, B. (1972b). Statistical methodology for nonperiodic cycles: from the covariance to r/s analysis. Ann. Econom. Soc. Measurement 1, 259{290.

[30] Mandelbrot, B. (1974). Intermittent turbulence in self-similar cascades; divergenceof high moments and dimension of the carrier. J. Fluid Mechanics 62, 331{358.

[31] Mandelbrot, B. (1975a). Fonctions al┬Âeatoires pluri-temporelles: approximation poissonienne du cas brownien et g┬Âeen┬Âeralisations. Comptes Rendus Acad. Sci. Paris 280A, 1075{1078.

[32] Mandelbrot, B. and M. Taylor (1967). On the distribution of stock price di®erences. Operations Research 15, 1057{1062.

[33] Mantegna, R. N. (1994). Fast, accurate algorithm for numerical simulation of L┬Âevy stable stochastic processes. Physical Review E 49 (5), 4677 { 4683.

[34] Nikias, C. L. and M. Shao (1995). Signal Processing with Alpha-Stable Distributions and Applications. New York: Wiley.

[35] Nolan, J. P. (2005a). Identi┬»cation of stable measures. In progress.

[36] Nolan, J. P. (2005b). Multivariate stable densities and distribution functions: general and elliptical case. Preprint, BundesBank November 2005 Conference.

[37] Nolan, J. P. (2006). Multivariate elliptically contoured stable distributions: theory and estimation. Submitted.

[38] Nolan, J. P. (2007). Metrics for multivariate stable distributions. Preprint.

[39] Nolan, J. P. (2008a, July). Advances in nonlinear signal processing for heavy tailed noise. IWAP 2008. 78

[40] Nolan, J. P. (2008b). Multivariate stable cumulative probabilities in polar form and related functions. In progress.

[41] Nolan, J. P. (2009). Stable Distributions - Models for Heavy Tailed Data. Boston: Birkh├äauser. Un┬»nished manuscript, Chapter 1 online at academic2.american.edu/»jpnolan.

[42] Nolan, J. P. and D. Ojeda (2006). Linear regression with general stable errors. Submitted.

[43] Nolan, J. P., A. Panorska, and J. H. McCulloch (2001). Estimation of stable spectral measures. Mathematical and Computer Modelling 34, 1113{1122.

[44] Nolan, J. P. and A. K. Panorska (1997). Data analysis for heavy tailed multivariate samples. Comm. in Stat. - Stochastic Models 13, 687{702.

[45] Nolan, J. P. and B. Rajput (1995). Calculation of multidimensional stable densities. Commun. Statist. - Simula. 24, 551{556.

[46] Nolan, J. P. and N. Ravishanker (2009). Simultaneous prediction intervals for ARMA proceses with stable time innovations. J. of Forecasting.

[47] Nolan, J. P. and A. Swami (Eds.) (1999). Proceedings of the ASA-IMS Conference on Heavy Tailed Distributions, Washington, DC

[48] _ Peach, G. (1981). "Theory of the pressure broadening and shift of spectral lines (http://journalsonline.tandf.co.uk/openurl.asp? genre=article&eissn=1460- 6976&volume=30&issue=3&spage=367)". Advances in Physics 30 (3): 367-474.

[49] _ V.M. Zolotarev (1986). One-dimensional Stable Distributions. American Mathematical Society.

[50] Samorodnitsky, G. (1988). Extrema of skewed stable processes. Stoch. Proc.Appl. 30, 17{39.

[51] Samorodnitsky, G. (1991). Probability tails of Gaussian extrema. Stoch. Proc.Appl. 38, 55{84.

[52] Samorodnitsky, G. (1993a). Integrability of stable processes. Probab. Math. Stat.. To appear.

[53] Samorodnitsky, G. (1993b). Possible sample paths of ®-stable self-similar pro- cesses. Stat. Probab. Letters. To appear.

[54] Samorodnitsky, G. (1995a). Association of in┬»nitely divisible random vectors. Stochastic Processes and Their Applications 55, 45{56.

[55] Stuck, B. W. (1976). Distinguishing stable probability measures. Bell System Technical Journal 55, 1125{1182.

[56] Stuck, B. W. and B. Kleiner (1974). A statistical analysis of telephone noise. Bell Syst. Tech. J. 53, 1263{1320

[57] Taqqu, M. (1978). A representation for self-similar processes. Stochastic Proc.and their Applic. 7, 55{64.

[58] Tsakalides, P. (1995, December). Array Signal Processing with Alpha-Stable Distributions. Ph. D. thesis, University Southern California.

[59] Tsakalides, P. and C. Nikias (1995). Maximum likelihood localization of sources in noise modeled as a stable processes. IEEE Trans. on Signal Proc. 43, 2700{2713.

[60] Tsakalides, P. and C. L. Nikias (1999). Robust space-time adaptive processing (stap) in non-gaussian clutter environments. IEE Proceedings: Radar, Sonar and Navigation 146 (2), 84{93.

[61] Tsakalides, P., R. Raspanti, and C. L. Nikias (1999). Angle/doppler estimation in heavy-tailed clutter backgrounds. IEEE Transactions on Aerospace and Electronic Systems 35 (2), 419436.

[62] Voit Johannes (2003). The Statistical Mechanics of Financial Markets (Texts and Monographs in Physics). Springer-Verlag. ISBN 3-540-00978-

[63] Zolotarev, V. M. (1986a). Asymptotic behavior of the Gaussian measure in l2. J. Soviet Math. 24, 2330{2334.

[64] Zolotarev, V. M. (1986b). One-dimensional Stable Distributions, Volume 65 of Translations of mathematical monographs. American Mathematical Society. Translation from the original 1983 Russian edition.

[65] Zolotarev, V. M. (1995). On representation of densities of stable laws by special functions. Theory Probab. Appl. 39, 354{362.