Applications of Stable Distributions in Time Series Analysis, Computer Sciences and Financial Markets
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.
[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.
BirkhÄauser.
[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: BirkhÄauser. 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.
[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.
BirkhÄauser.
[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: BirkhÄauser. 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.
@article{"International Journal of Business, Human and Social Sciences:60641", author = "Mohammad Ali Baradaran Ghahfarokhi and Parvin Baradaran Ghahfarokhi", title = "Applications of Stable Distributions in Time Series Analysis, Computer Sciences and Financial Markets", 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 = "stable distribution, SaS, infinite variance, heavy tail networks, VaR.", volume = "3", number = "1", pages = "110-5", }
