×

On \(1/f\) noise. (English) Zbl 1264.94060

Summary: Due to the fact that \(1/f\) noise gains the increasing interests in the field of biomedical signal processing and living systems, we present this introductive survey that may suffice to exhibit the elementary and the particularities of \(1/f\) noise in comparison with conventional random functions. Three theorems are given for highlighting the particularities of \(1/f\) noise. The first says that a random function with long-range dependence (LRD) is a \(1/f\) noise. The secondindicates that a heavy-tailed random function is in the class of \(1/f\) noise. The third provides a type of stochastic differential equations that produce \(1/f\) noise.

MSC:

94A12 Signal theory (characterization, reconstruction, filtering, etc.)
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M15 Inference from stochastic processes and spectral analysis
60G22 Fractional processes, including fractional Brownian motion

Software:

LSD; longmemo
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] W. Schottky, “Uber spontane Stromschwankungen in verschiedenen Elektri-zittsleitern,” Annalen der Physik, vol. 362, no. 23, pp. 541-567, 1918.
[2] W. Schottky, “Zur Berechnung und Beurteilung des Schroteffektes,” Annalen der Physik, vol. 373, no. 10, pp. 157-176, 1922.
[3] J. B. Johnson, “The Schottky effect in low frequency circuits,” Physical Review, vol. 26, no. 1, pp. 71-85, 1925.
[4] B. B. Mandelbrot, Multifractals and 1/f Noise, Springer, New York, NY, USA, 1998. · Zbl 1052.28500
[5] K. Fraedrich, U. Luksch, and R. Blender, “1/f model for long-time memory of the ocean surface temperature,” Physical Review E, vol. 70, no. 3, Article ID 037301, 4 pages, 2004.
[6] E. J. Wagenmakers, S. Farrell, and R. Ratcliff, “Estimation and interpretation of 1/f\alpha noise in human cognition,” Psychonomic Bulletin and Review, vol. 11, no. 4, pp. 579-615, 2004.
[7] F. Principato and G. Ferrante, “1/f noise decomposition in random telegraph signals using the wavelet transform,” Physica A, vol. 380, no. 1-2, pp. 75-97, 2007.
[8] V. P. Koverda and V. N. Skokov, “Maximum entropy in a nonlinear system with a 1/f power spectrum,” Physica A, vol. 391, no. 1-2, pp. 21-28, 2012. · Zbl 1274.60174
[9] Y. Nemirovsky, D. Corcos, I. Brouk, A. Nemirovsky, and S. Chaudhry, “1/f noise in advanced CMOS transistors,” IEEE Instrumentation and Measurement Magazine, vol. 14, no. 1, pp. 14-22, 2011.
[10] O. Miramontes and P. Rohani, “Estimating 1/f\alpha scaling exponents from short time-series,” Physica D, vol. 166, no. 3-4, pp. 147-154, 2002. · Zbl 0991.62068
[11] C. M. van Vliet, “Random walk and 1/f noise,” Physica A, vol. 303, no. 3-4, pp. 421-426, 2002. · Zbl 0983.60041
[12] J. S. Kim, Y. S. Kim, H. S. Min, and Y. J. Park, “Theory of 1/f noise currents in semiconductor devices with one-dimensional geometry and its application to Si Schottky barrier diodes,” IEEE Transactions on Electron Devices, vol. 48, no. 12, pp. 2875-2883, 2001.
[13] T. Antal, M. Droz, G. Györgyi, and Z. Rácz, “1/f noise and extreme value statistics,” Physical Review Letters, vol. 87, no. 24, Article ID 240601, 4 pages, 2001.
[14] B. Pilgram and D. T. Kaplan, “A comparison of estimators for 1/f noise,” Physica D, vol. 114, no. 1-2, pp. 108-122, 1998. · Zbl 0983.37103
[15] H. J. Jensen, “Lattice gas as a model of 1/f noise,” Physical Review Letters, vol. 64, no. 26, pp. 3103-3106, 1990.
[16] E. Marinari, G. Parisi, D. Ruelle, and P. Windey, “Random walk in a random environment and 1/f noise,” Physical Review Letters, vol. 50, no. 17, pp. 1223-1225, 1983. · Zbl 0532.60062
[17] F. N. Hooge, “1/f noise,” Physica B, vol. 83, no. 1, pp. 14-23, 1976.
[18] M. B. Weissman, “Simple model for 1/f noise,” Physical Review Letters, vol. 35, no. 11, pp. 689-692, 1975.
[19] F. N. Hooge, “Discussion of recent experiments on 1/f noise,” Physica, vol. 60, no. 1, pp. 130-144, 1972.
[20] C. Cattani, “Harmonic wavelet approximation of random, fractal and high frequency signals,” Telecommunication Systems, vol. 43, no. 3-4, pp. 207-217, 2010. · Zbl 05803252
[21] C. Cattani, “On the existence of wavelet symmetries in archaea DNA,” Computational and Mathematical Methods in Medicine, vol. 2012, Article ID 673934, 21 pages, 2012. · Zbl 1234.92014
[22] C. Cattani, E. Laserra, and I. Bochicchio, “Simplicial approach to fractal structures,” Mathematical Problems in Engineering, vol. 2012, Article ID 958101, 21 pages, 2012. · Zbl 1264.28005
[23] C. Cattani, “Fractional calculus and Shannon wavelet,” Mathematical Problems in Engineering, vol. 2012, Article ID 502812, 26 pages, 2012. · Zbl 1264.42016
[24] C. Cattani, G. Pierro, and G. Altieri, “Entropy and multifractality for the mye-loma multiple TET 2 gene,” Mathematical Problems in Engineering, vol. 2012, Article ID 193761, 14 pages, 2012. · Zbl 1264.92017
[25] M. S. Keshner, “1/f noise,” Proceedings of the IEEE, vol. 70, no. 3, pp. 212-218, 1982.
[26] B. Ninness, “Estimation of 1/f Noise,” IEEE Transactions on Information Theory, vol. 44, no. 1, pp. 32-46, 1998. · Zbl 0905.94009
[27] B. Yazici and R. L. Kashyap, “A class of second-order stationary self-similar processes for 1/f phenomena,” IEEE Transactions on Signal Processing, vol. 45, no. 2, pp. 396-410, 1997.
[28] G. W. Wornell, “Wavelet-based representations for the 1/f family of fractal processes,” Proceedings of the IEEE, vol. 81, no. 10, pp. 1428-1450, 1993.
[29] B. B. Mandelbrot, “Some noises with 1/f spectrum, a bridge between direct current and white noise,” IEEE Transactions on Information Theory, vol. 13, no. 2, pp. 289-298, 1967. · Zbl 0148.40507
[30] N. J. Kasdin, “Discrete simulation of colored noise and stochastic processes and 1/f\alpha power law noise generation,” Proceedings of the IEEE, vol. 83, no. 5, pp. 802-827, 1995.
[31] G. Corsini and R. Saletti, “1/f\gamma power spectrum noise sequence generator,” IEEE Transactions on Instrumentation and Measurement, vol. 37, no. 4, pp. 615-619, 1988.
[32] W. T. Li and D. Holste, “Universal 1/f noise, crossovers of scaling exponents, and chromosome-specific patterns of guanine-cytosine content in DNA sequences of the human genome,” Physical Review E, vol. 71, no. 4, Article ID 041910, 9 pages, 2005.
[33] W. T. Li, G. Stolovitzky, P. Bernaola-Galván, and J. L. Oliver, “Compositional heterogeneity within, and uniformity between, DNA sequences of yeast chromosomes,” Genome Research, vol. 8, no. 9, pp. 916-928, 1998.
[34] W. T. Li and K. Kaneko, “Long-range correlation and partial spectrum in a noncoding DNA sequence,” Europhysics Letters, vol. 17, no. 7, pp. 655-660, 1992.
[35] P. C. Ivanov, L. A. Nunes Amaral, A. L. Goldberger et al., “From 1/f noise to multifractal cascades in heartbeat dynamics,” Chaos, vol. 11, no. 3, pp. 641-652, 2001. · Zbl 0990.92024
[36] R. N. Mantegna and H. E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press, Cambridge, UK, 2000. · Zbl 1138.91300
[37] W. Q. Duan and H. E. Stanley, “Cross-correlation and the predictability of financial return series,” Physica A, vol. 390, no. 2, pp. 290-296, 2010.
[38] B. Podobnik, D. Horvatic, A. Lam Ng, H. E. Stanley, and P. C. Ivanov, “Modeling long-range cross-correlations in two-component ARFIMA and FIARCH processes,” Physica A, vol. 387, no. 15, pp. 3954-3959, 2008.
[39] G. Aquino, M. Bologna, P. Grigolini, and B. J. West, “Beyond the death of linear response: 1/f optimal information transport,” Physical Review Letters, vol. 105, no. 4, Article ID 040601, 4 pages, 2010.
[40] B. J. West and P. Grigolini, “Chipping away at memory,” Biological Cybernetics, vol. 103, no. 2, pp. 167-174, 2010. · Zbl 1266.92047
[41] B. J. West and M. F. Shlesinger, “On the ubiquity of 1/f noise,” International Journal of Modern Physics B, vol. 3, no. 6, pp. 795-819, 1989.
[42] A. L. Goldberger, V. Bhargava, B. J. West, and A. J. Mandell, “On the mechanism of cardiac electrical stability. The fractal hypothesis,” Biophysical Journal, vol. 48, no. 3, pp. 525-528, 1985.
[43] T. Musha, H. Takeuchi, and T. Inoue, “1/f fluctuations in the spontaneous spike discharge intervals of a giant snail neuron,” IEEE Transactions on Biomedical Engineering, vol. 30, no. 3, pp. 194-197, 1983.
[44] M. Kobayashi and T. Musha, “1/f fluctuation of heartbeat period,” IEEE Transactions on Biomedical Engineering, vol. 29, no. 6, pp. 456-457, 1982.
[45] B. Neumcke, “1/f noise in membranes,” Biophysics of Structure and Mechanism, vol. 4, no. 3, pp. 179-199, 1978.
[46] J. R. Clay and M. F. Shlesinger, “Unified theory of 1/f and conductance noise in nerve membrane,” Journal of Theoretical Biology, vol. 66, no. 4, pp. 763-773, 1977.
[47] E. Frehland, “Diffusion as a source of 1/f noise,” The Journal of Membrane Biology, vol. 32, no. 1, pp. 195-196, 1977.
[48] M. E. Green, “Diffusion and 1/f noise,” The Journal of Membrane Biology, vol. 28, no. 1, pp. 181-186, 1976.
[49] I. Csabai, “1/f noise in computer network traffic,” Journal of Physics A, vol. 27, no. 12, pp. L417-L421, 1994.
[50] M. Takayasu, H. Takayasu, and T. Sato, “Critical behaviors and 1/f noise in information traffic,” Physica A, vol. 233, no. 3-4, pp. 824-834, 1996.
[51] V. Paxson and S. Floyd, “Wide area traffic: the failure of Poisson modeling,” IEEE/ACM Transactions on Networking, vol. 3, no. 3, pp. 226-244, 1995.
[52] W. Willinger, R. Govindan, S. Jamin, V. Paxson, and S. Shenker, “Scaling phenomena in the internet: critically examining criticality,” Proceedings of the National Academy of Sciences of the United States of America, vol. 99, supplement 1, pp. 2573-2580, 2002.
[53] P. Loiseau, P. Gon\ccalves, G. Dewaele, P. Borgnat, P. Abry, and P. V. B. Primet, “Investigating self-similarity and heavy-tailed distributions on a large-scale experimental facility,” IEEE/ACM Transactions on Networking, vol. 18, no. 4, pp. 1261-1274, 2010. · Zbl 05890442
[54] P. Abry, P. Borgnat, F. Ricciato, A. Scherrer, and D. Veitch, “Revisiting an old friend: on the observability of the relation between long range dependence and heavy tail,” Telecommunication Systems, vol. 43, no. 3-4, pp. 147-165, 2010. · Zbl 05803248
[55] A. M. Yaglom, Correlation Theory of Stationary and Related Random Functions, vol. 1, Springer, New York, NY, USA, 1987. · Zbl 0685.62077
[56] A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York, NY, USA, 1997.
[57] B. W. Lindgren and G. W. McElrath, Introduction to Probability and Statistics, The Macmillan, New York, NY, USA, 1959. · Zbl 0085.34602
[58] J. L. Doob, “The elementary Gaussian processes,” Annals of Mathematical Statistics, vol. 15, pp. 229-282, 1944. · Zbl 0060.28907
[59] A. N. Kolmogorov, Fundamental of Probability, Business Press, Shanghai, China, 1954, Translated from Russian by S.-T. Ding.
[60] G. E. P. Box, G. M. Jenkins, and G. C. Reinsel, Time Series Analysis: Forecasting and Control, Prentice Hall, Englewood Cliffs, NJ, USA, 1994. · Zbl 0858.62072
[61] J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedure, Wiley Series in Probability and Statistics, John Wiley & Sons, Hoboken, NJ, USA, 3rd edition, 2000. · Zbl 0953.62128
[62] W. A. Fuller, Introduction to Statistical Time Series, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, New York, NY, USA, 2nd edition, 1996. · Zbl 0851.62057
[63] J. Beran, Statistics for Long-Memory Processes, vol. 61 of Monographs on Statistics and Applied Probability, Chapman & Hall, New York, NY, USA, 1994. · Zbl 0869.60045
[64] J. Beran, “Statistical methods for data with long-range dependence,” Statistical Science, vol. 7, no. 4, pp. 404-416, 1992.
[65] A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 2, MIT Press, Cambridge, Mass, USA, 1971. · Zbl 1140.76003
[66] A. N. Kolmogorov, “Local structure of turbulence in an incompressible viscous fluid at very high Reynolds numbers,” Soviet Physics Uspekhi, vol. 10, no. 6, pp. 734-736, 1968.
[67] N. C. Nigam, Introduction to Random Vibrations, MIT Press, Cambridge, Mass, USA, 1983.
[68] T. T. Song and M. Grigoriu, Random Vibration of Mechanical and Structural Systems, Prentice Hall, New York, NY, USA, 1993. · Zbl 0788.73005
[69] C. M. Harris, Shock and Vibration Handbook, McGraw-Hill, New York, NY, USA, 4th edition, 1995.
[70] H. Czichos, T. Saito, and L. Smith, Springer Handbook of Metrology and Testing, Springer, New York, NY, USA, 2011.
[71] W. N. Sharpe Jr., Springer Handbook of Experimental Solid Mechanics, Springer, New York, NY, USA, 2008.
[72] C. Tropea, A. L. Yarin, and J. F. Foss, Eds., Springer Handbook of Experimental Fluid Mechanics, Springer, New York, NY, USA, 2007.
[73] K. H. Grote and E. K. Antonsson, Eds., Springer Handbook of Mechanical Engineering, Springer, New York, NY, USA, 2009.
[74] R. Kramme, K. P. Hoffmann, and R. S. Pozos, Springer Handbook of Medical Technology, Springer, New York, NY, USA, 2012.
[75] W. Kresse and D. M. Danko, Springer Handbook of Geographic Information, Springer, New York, NY, USA, 2012.
[76] S. S. Bhattacharyya, F. Deprettere, R. Leupers, and J. Takala, Eds., Handbook of Signal Processing Systems, Springer, New York, NY, USA, 2010. · Zbl 1285.94001
[77] S. K. Mitra and J. F. Kaiser, Handbook for Digital Signal Processing, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0832.94001
[78] W. A. Woyczyński, A First Course in Statistics for Signal Analysis, Birkhäuser, Boston, Mass, USA, 2006. · Zbl 1269.62007
[79] R. A. Bailey, Design of Comparative Experiments, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, UK, 2008. · Zbl 1155.62054
[80] ASME, Measurement Uncertainty Part 1, Instruments and Apparatus, Sup-plement to ASME, Performance Test Codes, ASME, New York, NY, USA, 1986.
[81] D. Sheskin, Statistical Tests and Experimental Design: A Guidebook, Gardner Press, New York, NY, USA, 1984.
[82] T. W. MacFarland, Two-Way Analysis of Variance, Springer, New York, NY, USA, 2012. · Zbl 1304.62010
[83] A. K. Gupta, W. B. Zeng, and Y. Wu, Probability and Statistical Models: Foundations for Problems in Reliability and Financial Mathematics, Birkhäuser, Boston, Mass, USA, 2010. · Zbl 1215.60001
[84] P. Fieguth, Statistical Image Processing and Multidimensional Modeling, Information Science and Statistics, Springer, New York, NY, USA, 2011. · Zbl 1209.94001
[85] J. Nauta, Statistics in Clinical Vaccine Trials, Springer, New York, NY, USA, 2011. · Zbl 1275.62024
[86] A. Gelman, “Analysis of variance-why it is more important than ever,” The Annals of Statistics, vol. 33, no. 1, pp. 1-53, 2005. · Zbl 1064.62082
[87] C. G. Pendse, “A note on mathematical expectation,” The Mathematical Gazette, vol. 22, no. 251, pp. 399-402, 1938. · JFM 64.1190.02
[88] B. B. Mandelbrot, Gaussian Self-Affinity and Fractals, Springer, New York, NY, USA, 2001.
[89] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Stochastic Modeling, Chapman & Hall, New York, NY, USA, 1994. · Zbl 0925.60027
[90] J. Beran, R. Sherman, M. S. Taqqu, and W. Willinger, “Long-range dependence in variable-bit-rate video traffic,” IEEE Transactions on Communications, vol. 43, no. 234, pp. 1566-1579, 1995.
[91] I. M. Gelfand and K. Vilenkin, Generalized Functions, vol. 1, Academic Press, New York, NY, USA, 1964.
[92] M. Li and S. C. Lim, “Power spectrum of generalized Cauchy process,” Telecommunication Systems, vol. 43, no. 3-4, pp. 219-222, 2010. · Zbl 05803253
[93] M. Li and S. C. Lim, “A rigorous derivation of power spectrum of fractional Gaussian noise,” Fluctuation and Noise Letters, vol. 6, no. 4, pp. C33-C36, 2006.
[94] B. B. Mandelbrot and J. W. van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Review, vol. 10, pp. 422-437, 1968. · Zbl 0179.47801
[95] P. Flandrin, “On the spectrum of fractional Brownian motions,” IEEE Transactions on Information Theory, vol. 35, no. 1, pp. 197-199, 1989.
[96] M. Li, “Fractal time series-a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010. · Zbl 1191.37002
[97] S. V. Muniandy and S. C. Lim, “Modeling of locally self-similar processes using multifractional Brownian motion of Riemann-Liouville type,” Physical Review E, vol. 63, no. 4, Article ID 046104, 7 pages, 2001. · Zbl 1029.60026
[98] V. M. Sithi and S. C. Lim, “On the spectra of Riemann-Liouville fractional Brownian motion,” Journal of Physics A, vol. 28, no. 11, pp. 2995-3003, 1995. · Zbl 0828.60099
[99] S. C. Lim and S. V. Muniandy, “On some possible generalizations of fractional Brownian motion,” Physics Letters A, vol. 266, no. 2-3, pp. 140-145, 2000. · Zbl 1068.82518
[100] J. P. Chilès and P. Delfiner, Geostatistics, Modeling Spatial Uncertainty, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley & Sons, New York, NY, USA, 1999. · Zbl 0922.62098
[101] M. Li, C. Cattani, and S. Y. Chen, “Viewing sea level by a one-dimensional random function with long memory,” Mathematical Problems in Engineering, vol. 2011, Article ID 654284, 13 pages, 2011. · Zbl 05829261
[102] S. C. Lim and M. Li, “A generalized Cauchy process and its application to relaxation phenomena,” Journal of Physics A, vol. 39, no. 12, pp. 2935-2951, 2006. · Zbl 1090.82013
[103] S. C. Lim and L. P. Teo, “Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure,” Stochastic Processes and Their Applications, vol. 119, no. 4, pp. 1325-1356, 2009. · Zbl 1161.60314
[104] M. Li and S. C. Lim, “Modeling network traffic using generalized Cauchy process,” Physica A, vol. 387, no. 11, pp. 2584-2594, 2008.
[105] P. Vengadesh, S. V. Muniandy, and W. H. A. Majid, “Fractal morphological analysis of bacteriorhodopsin (bR) layers deposited onto indium tin oxide (ITO) electrodes,” Materials Science and Engineering C, vol. 29, no. 5, pp. 1621-1626, 2009.
[106] R. J. Martin and A. M. Walker, “A power-law model and other models for long-range dependence,” Journal of Applied Probability, vol. 34, no. 3, pp. 657-670, 1997. · Zbl 0882.62085
[107] R. J. Martin and J. A. Eccleston, “A new model for slowly-decaying correlations,” Statistics and Probability Letters, vol. 13, no. 2, pp. 139-145, 1992. · Zbl 0752.60031
[108] W. A. Woodward, Q. C. Cheng, and H. L. Gray, “A k-factor GARMA long-memory model,” Journal of Time Series Analysis, vol. 19, no. 4, pp. 485-504, 1998. · Zbl 1017.62083
[109] C. Ma, “Power-law correlations and other models with long-range dependence on a lattice,” Journal of Applied Probability, vol. 40, no. 3, pp. 690-703, 2003. · Zbl 1041.60046
[110] C. Ma, “A class of stationary random fields with a simple correlation structure,” Journal of Multivariate Analysis, vol. 94, no. 2, pp. 313-327, 2005. · Zbl 1070.62084
[111] M. Li, W. Jia, and W. Zhao, “Correlation form of timestamp increment sequences of self-similar traffic on Ethernet,” Electronics Letters, vol. 36, no. 19, pp. 1668-1669, 2000.
[112] M. Li and W. Zhao, “Quantitatively investigating locally weak stationarity of modified multifractional Gaussian noise,” Physica A, vol. 391, no. 24, pp. 6268-6278, 2012.
[113] E. G. Tsionas, “Estimating multivariate heavy tails and principal directions easily, with an application to international exchange rates,” Statistics and Probability Letters, vol. 82, no. 11, pp. 1986-1989, 2012. · Zbl 1312.62116
[114] J. Lin, “Second order asymptotics for ruin probabilities in a renewal risk model with heavy-tailed claims,” Insurance: Mathematics and Economics, vol. 51, no. 2, pp. 422-429, 2012. · Zbl 1284.91251
[115] K. Yu, M. L. Huang, and P. H. Brill, “An algorithm for fitting heavy-tailed distributions via generalized hyperexponentials,” INFORMS Journal on Computing, vol. 24, no. 1, pp. 42-52, 2012. · Zbl 1465.62043
[116] R. Luger, “Finite-sample bootstrap inference in GARCH models with heavy-tailed innovations,” Computational Statistics and Data Analysis, vol. 56, no. 11, pp. 3198-3211, 2012. · Zbl 1254.91679
[117] T. Ishihara and Y. Omori, “Efficient Bayesian estimation of a multivariate stochastic volatility model with cross leverage and heavy-tailed errors,” Computational Statistics and Data Analysis, 2010. · Zbl 1255.62066
[118] J. Diebolt, L. Gardes, S. Girard, and A. Guillou, “Bias-reduced extreme quantile estimators of Weibull tail-distributions,” Journal of Statistical Planning and Inference, vol. 138, no. 5, pp. 1389-1401, 2008. · Zbl 1250.62024
[119] J. Beran, B. Das, and D. Schell, “On robust tail index estimation for linear long-memory processes,” Journal of Time Series Analysis, vol. 33, no. 3, pp. 406-423, 2012. · Zbl 1301.62080
[120] P. Barbe and W. P. McCormick, “Heavy-traffic approximations for fractionally integrated random walks in the domain of attraction of a non-Gaussian stable distribution,” Stochastic Processes and Their Applications, vol. 122, no. 4, pp. 1276-1303, 2012. · Zbl 1254.60035
[121] C. Weng and Y. Zhang, “Characterization of multivariate heavy-tailed distribution families via copula,” Journal of Multivariate Analysis, vol. 106, pp. 178-186, 2012. · Zbl 1236.62048
[122] C. B. García, J. García Pérez, and J. R. van Dorp, “Modeling heavy-tailed, skewed and peaked uncertainty phenomena with bounded support,” Statistical Methods and Applications, vol. 20, no. 4, pp. 146-166, 2011. · Zbl 1238.62012
[123] V. H. Lachos, T. Angolini, and C. A. Abanto-Valle, “On estimation and local influence analysis for measurement errors models under heavy-tailed distributions,” Statistical Papers, vol. 52, no. 3, pp. 567-590, 2011. · Zbl 1434.62152
[124] V. Ganti, K. M. Straub, E. Foufoula-Georgiou, and C. Paola, “Space-time dynamics of depositional systems: experimental evidence and theoretical modeling of heavy-tailed statistics,” Journal of Geophysical Research F: Earth Surface, vol. 116, no. 2, Article ID F02011, 17 pages, 2011.
[125] U. J. Dixit and M. J. Nooghabi, “Efficient estimation in the Pareto distribution with the presence of outliers,” Statistical Methodology, vol. 8, no. 4, pp. 340-355, 2011. · Zbl 1215.62021
[126] P. Nándori, “Recurrence properties of a special type of heavy-tailed random walk,” Journal of Statistical Physics, vol. 142, no. 2, pp. 342-355, 2011. · Zbl 1223.60021
[127] D. Ceresetti, G. Molinié, and J. D. Creutin, “Scaling properties of heavy rainfall at short duration: a regional analysis,” Water Resources Research, vol. 46, no. 9, Article ID W09531, 12 pages, 2010.
[128] A. Charpentier and A. Oulidi, “Beta kernel quantile estimators of heavy-tailed loss distributions,” Statistics and Computing, vol. 20, no. 1, pp. 35-55, 2010.
[129] P. Embrechts, J. Ne, and M. V. Wüthrich, “Additivity properties for Value-at-Risk under Archimedean dependence and heavy-tailedness,” Insurance: Mathematics and Economics, vol. 44, no. 2, pp. 164-169, 2009. · Zbl 1163.91431
[130] I. F. Alves, L. de Haan, and C. Neves, “A test procedure for detecting super-heavy tails,” Journal of Statistical Planning and Inference, vol. 139, no. 2, pp. 213-227, 2009. · Zbl 1149.62036
[131] J. Beirlant, E. Joossens, and J. Segers, “Second-order refined peaks-over-threshold modelling for heavy-tailed distributions,” Journal of Statistical Planning and Inference, vol. 139, no. 8, pp. 2800-2815, 2009. · Zbl 1162.62044
[132] R. Ibragimov, “Heavy-tailedness and threshold sex determination,” Statistics and Probability Letters, vol. 78, no. 16, pp. 2804-2810, 2008. · Zbl 1154.62084
[133] R. Delgado, “A reflected fBm limit for fluid models with ON/OFF sources under heavy traffic,” Stochastic Processes and Their Applications, vol. 117, no. 2, pp. 188-201, 2007. · Zbl 1108.60074
[134] M. S. Taqqu, “The modelling of ethernet data and of signals that are heavy-tailed with infinite variance,” Scandinavian Journal of Statistics, vol. 29, no. 2, pp. 273-295, 2002. · Zbl 1020.60082
[135] B. G. Lindsay, J. Kettenring, and D. O. Siegmund, “A report on the future of statistics,” Statistical Science, vol. 19, no. 3, pp. 387-413, 2004. · Zbl 1100.62516
[136] S. Resnick, “On the foundations of multivariate heavy-tail analysis,” Journal of Applied Probability, vol. 41, pp. 191-212, 2004. · Zbl 1049.62056
[137] S. Resnick and H. Rootzén, “Self-similar communication models and very heavy tails,” The Annals of Applied Probability, vol. 10, no. 3, pp. 753-778, 2000. · Zbl 1083.60521
[138] J. Cai and Q. Tang, “On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications,” Journal of Applied Probability, vol. 41, no. 1, pp. 117-130, 2004. · Zbl 1054.60012
[139] V. Limic, “A LIFO queue in heavy traffic,” The Annals of Applied Probability, vol. 11, no. 2, pp. 301-331, 2001. · Zbl 1015.60079
[140] H. Le and A. O. ’Hagan, “A class of bivariate heavy-tailed distributions,” San-Khy\Ba: The Indian Journal of Statistics, Series B, vol. 60, no. 1, pp. 82-100, 1998. · Zbl 0972.62017
[141] M. C. Bryson, “Heavy-tailed distributions: properties and tests,” Technometrics, vol. 16, no. 1, pp. 61-68, 1974. · Zbl 0337.62065
[142] J. Beran, “Discussion: heavy tail modeling and teletraffic data,” The Annals of Statistics, vol. 25, no. 5, pp. 1852-1856, 1997. · Zbl 0942.62097
[143] S. Ahn, J. H. T. Kim, and V. Ramaswami, “A new class of models for heavy tailed distributions in finance and insurance risk,” Insurance: Mathematics and Economics, vol. 51, no. 1, pp. 43-52, 2012. · Zbl 1284.60024
[144] V. Pisarenko and M. Rodkin, Heavy-Tailed Distributions in Disaster Analysis, Springer, New York, NY, USA, 2010. · Zbl 1352.86015
[145] S. I. Resnick, Heavy-Tail Phenomena Probabilistic and Statistical Modeling, Springer, New York, NY, USA, 2007, Probabilistic and statistical modeling. · Zbl 1152.62029
[146] R. J. Adler, R. E. Feldman, and M. S. Taqqu, Eds., A Practical Guide to Heavy Tails: Statistical Techniques and Applications,, Birkhäuser, Boston, Mass, USA, 1998. · Zbl 0901.00010
[147] M. Li and W. Zhao, “Visiting power laws in cyber-physical networking systems,” Mathematical Problems in Engineering, vol. 2012, Article ID 302786, 13 pages, 2012. · Zbl 06173174
[148] L. Xu, P. C. Ivanov, K. Hu, Z. Chen, A. Carbone, and H. E. Stanley, “Quantifying signals with power-law correlations: a comparative study of detrended fluctuation analysis and detrended moving average techniques,” Physical Review E, vol. 71, no. 5, Article ID 051101, 14 pages, 2005.
[149] M. Li and J. Y. Li, “On the predictability of long-range dependent series,” Mathematical Problems in Engineering, vol. 2010, Article ID 397454, 9 pages, 2010. · Zbl 1191.62160
[150] W. Hürlimann, “From the general affine transform family to a Pareto type IV model,” Journal of Probability and Statistics, vol. 2009, Article ID 364901, 10 pages, 2009. · Zbl 1201.62031
[151] A. André, “Limit theorems for randomly selected adjacent order statistics from a Pareto distribution,” International Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 21, pp. 3427-3441, 2005. · Zbl 1087.60020
[152] H. E. Stanley, “Power laws and universality,” Nature, vol. 378, no. 6557, p. 554, 1995.
[153] I. Eliazar and J. Klafter, “A probabilistic walk up power laws,” Physics Reports, vol. 511, no. 3, pp. 143-175, 2012.
[154] A. R. Bansal, G. Gabriel, and V. P. Dimri, “Power law distribution of susceptibility and density and its relation to seismic properties: an example from the German Continental Deep Drilling Program (KTB),” Journal of Applied Geophysics, vol. 72, no. 2, pp. 123-128, 2010.
[155] S. Milojević, “Power law distributions in information science: making the case for logarithmic binning,” Journal of the American Society for Information Science and Technology, vol. 61, no. 12, pp. 2417-2425, 2010.
[156] Y. Wu, Q. Ye, J. Xiao, and L. X. Li, “Modeling and statistical properties of human view and reply behavior in on-line society,” Mathematical Problems in Engineering, vol. 2012, Article ID 969087, 7 pages, 2012. · Zbl 1264.91098
[157] A. Fujihara, M. Uchida, and H. Miwa, “Universal power laws in the threshold network model: a theoretical analysis based on extreme value theory,” Physica A, vol. 389, no. 5, pp. 1124-1130, 2010.
[158] A. Saiz, “Boltzmann power laws,” Physica A, vol. 389, no. 2, pp. 225-236, 2010.
[159] A. Jaishankar and G. H. McKinley, “Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations,” Proceedings of the Royal Society of London Series A, vol. 469, no. 2149, Article ID 20120284, 2013. · Zbl 1371.74066
[160] X. Zhao, P. J. Shang, and Y. L. Pang, “Power law and stretched exponential effects of extreme events in Chinese stock markets,” Fluctuation and Noise Letters, vol. 9, no. 2, pp. 203-217, 2010.
[161] P. Kokoszka and T. Mikosch, “The integrated periodogram for long-memory processes with finite or infinite variance,” Stochastic Processes and Their Applications, vol. 66, no. 1, pp. 55-78, 1997. · Zbl 0885.62108
[162] D. Belomestny, “Spectral estimation of the Lévy density in partially observed affine models,” Stochastic Processes and Their Applications, vol. 121, no. 6, pp. 1217-1244, 2011. · Zbl 1216.62132
[163] T. Simon, “Fonctions de Mittag-Leffler et processus de Lévy stables sans sauts négatifs,” Expositiones Mathematicae, vol. 28, no. 3, pp. 290-298, 2010. · Zbl 1194.33022
[164] R. Lambiotte and L. Brenig, “Truncated Lévy distributions in an inelastic gas,” Physics Letters A, vol. 345, no. 4-6, pp. 309-313, 2005. · Zbl 05314211
[165] G. Terdik, W. A. Woyczynski, and A. Piryatinska, “Fractional- and integer-order moments, and multiscaling for smoothly truncated Lévy flights,” Physics Letters A, vol. 348, no. 3-6, pp. 94-109, 2006.
[166] I. Koponen, “Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process,” Physical Review E, vol. 52, no. 1, pp. 1197-1199, 1995.
[167] J. Behboodian, A. Jamalizadeh, and N. Balakrishnan, “A new class of skew-Cauchydistributions,” Statistics and Probability Letters, vol. 76, no. 14, pp. 1488-1493, 2006. · Zbl 1096.60011
[168] P. Garbaczewski, “Cauchy flights in confining potentials,” Physica A, vol. 389, no. 5, pp. 936-944, 2010.
[169] A. J. Field, U. Harder, and P. G. Harrison, “Measurement and modelling of self-similar traffic in computer networks,” IEE Proceedings-Communications, vol. 151, no. 4, pp. 355-363, 2004.
[170] G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill, New York, NY, USA, 1961. · Zbl 0121.00103
[171] H. Konno and Y. Tamura, “A generalized Cauchy process having cubic non-linearity,” Reports on Mathematical Physics, vol. 67, no. 2, pp. 179-195, 2011. · Zbl 1235.62122
[172] H. Konno and F. Watanabe, “Maximum likelihood estimators for generalized Cauchy processes,” Journal of Mathematical Physics, vol. 48, no. 10, Article ID 103303, 19 pages, 2007. · Zbl 1152.81516
[173] I. A. Lubashevsky, “Truncated Lévy flights and generalized Cauchy processes,” European Physical Journal B, vol. 82, no. 2, pp. 189-195, 2011.
[174] Y. Liang and W. Chen, “A survey on computing Lévy stable distributions and a new MATLAB toolbox,” Signal Processing, vol. 93, no. 1, pp. 244-251, 2013.
[175] G. Terdik and T. Gyires, “Lévy flights and fractal modeling of internet traffic,” IEEE/ACM Transactions on Networking, vol. 17, no. 1, pp. 120-129, 2009.
[176] E. E. Kuruo\vglu, “Density parameter estimation of skewed \alpha -stable distributions,” IEEE Transactions on Signal Processing, vol. 49, no. 10, pp. 2192-2201, 2001. · Zbl 1369.62021
[177] A. P. Petropulu, J. C. Pesquet, X. Yang, and J. J. Yin, “Power-law shot noise and its relationship to long-memory \alpha -stable processes,” IEEE Transactions on Signal Processing, vol. 48, no. 7, pp. 1883-1892, 2000. · Zbl 0979.94020
[178] S. Cohen and G. Samorodnitsky, “Random rewards, fractional brownian local times and stable self-similar processes,” The Annals of Applied Probability, vol. 16, no. 3, pp. 1442-1461, 2006. · Zbl 1133.60016
[179] M. Shao and C. L. Nikias, “Signal processing with fractional lower order moments: stable processes and their applications,” Proceedings of the IEEE, vol. 81, no. 7, pp. 986-1010, 1993.
[180] L. Landau, “On the energy loss of fast particles by ionization,” Journal of Physics, vol. 8, pp. 201-205, 1944.
[181] D. H. Wilkinson, “Ionization energy loss by charged particles part I. The Landau distribution,” Nuclear Instruments and Methods in Physics Research A, vol. 383, no. 2-3, pp. 513-515, 1996.
[182] T. Tabata and R. Ito, “Approximations to Landau’s distribution functions for the ionization energy loss of fast electrons,” Nuclear Instruments and Methods, vol. 158, pp. 521-523, 1979.
[183] J. Holtsmark, “Uber die Verbreiterung von Spektrallinien,” Annalen der Physik, vol. 363, no. 7, pp. 577-630, 1919.
[184] B. Pittel, W. A. Woyczynski, and J. A. Mann, “Random tree-type partitions as a model for acyclic polymerization: holtsmark (3/2-stable) distribution of the supercritical gel,” The Annals of Probability, vol. 18, no. 1, pp. 319-341, 1990. · Zbl 0743.60110
[185] D. G. Hummer, “Rational approximations for the holtsmark distribution, its cumulative and derivative,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 36, no. 1, pp. 1-5, 1986.
[186] R. G. Garroppo, S. Giordano, M. Pagano, and G. Procissi, “Testing \alpha -stable processes in capturing the queuing behavior of broadband teletraffic,” Signal Processing, vol. 82, no. 12, pp. 1861-1872, 2002. · Zbl 1006.94026
[187] J. R. Gallardo, D. Makrakis, and L. Orozco-Barbosa, “Use of \alpha -stable self-similar stochastic processes for modeling traffic in broadband networks,” Performance Evaluation, vol. 40, no. 1, pp. 71-98, 2000. · Zbl 1052.68509
[188] A. Karasaridis and D. Hatzinakos, “Network heavy traffic modeling using \alpha -stable self-similar processes,” IEEE Transactions on Communications, vol. 49, no. 7, pp. 1203-1214, 2001. · Zbl 1063.68519
[189] P. R. de Montmort, “Essay d’analyse sur les jeux de hazard,” 1713. · Zbl 0476.60001
[190] P. R. de Montmort, “Essay d’analyse sur les jeux de hazard,” American Mathematical Society, 1980. · Zbl 0476.60001
[191] D. Bernoulli, “Exposi-tion of a new theory on the measurement of risk,” Econometrica, vol. 22, no. 1, pp. 22-36, 1954. · Zbl 0055.12004
[192] W. T. Coffey, Y. P. Kalmykov, and J. T. Waldron, The Langevin Equation, World Scientific, Singapore, 2nd edition, 2004. · Zbl 1098.82001
[193] S. F. Kwok, “Langevin equation with multiplicative white noise: transfor-mation of diffusion processes into the Wiener process in different prescrip-tions,” Annals of Physics, vol. 327, no. 8, pp. 1989-1997, 2012. · Zbl 1253.82067
[194] A. V. Medino, S. R. C. Lopes, R. Morgado, and C. C. Y. Dorea, “Generalized Langevin equation driven by Lévy processes: a probabilistic, numerical and time series based approach,” Physica A, vol. 391, no. 3, pp. 572-581, 2012.
[195] D. Panja, “Generalized langevin equation formulation for anomalous polymer dynamics,” Journal of Statistical Mechanics, vol. 2010, no. 2, Article ID L02001, 2010.
[196] A. Bazzani, G. Bassi, and G. Turchetti, “Diffusion and memory effects for stochastic processes and fractional Langevin equations,” Physica A, vol. 324, no. 3-4, pp. 530-550, 2003. · Zbl 1050.82029
[197] E. Lutz, “Fractional Langevin equation,” Physical Review E, vol. 64, no. 5, Article ID 051106, 4 pages, 2001. · Zbl 1308.82050
[198] M. G. McPhie, P. J. Daivis, I. K. Snook, J. Ennis, and D. J. Evans, “Generalized Langevin equation for nonequilibrium systems,” Physica A, vol. 299, no. 3-4, pp. 412-426, 2001. · Zbl 0972.82060
[199] K. S. Fa, “Fractional Langevin equation and Riemann-Liouville fractional derivative,” European Physical Journal E, vol. 24, no. 2, pp. 139-143, 2007.
[200] B. Ahmad, J. J. Nieto, A. Alsaedi, and M. El-Shahed, “A study of nonlinear Langevin equation involving two fractional orders in different intervals,” Non-Linear Analysis: Real World Applications, vol. 13, no. 2, pp. 599-606, 2012. · Zbl 1238.34008
[201] B. Ahmad and J. J. Nieto, “Solvability of nonlinear Langevin equation involving two fractional orders with Dirichlet boundary conditions,” International Journal of Differential Equations, vol. 2012, Article ID 649486, 10 pages, 2010. · Zbl 1207.34007
[202] S. C. Kou and X. S. Xie, “Generalized langevin equation with fractional gaussian noise: subdiffusion within a single protein molecule,” Physical Review Letters, vol. 93, no. 18, Article ID 180603, 4 pages, 2004.
[203] H. C. Fogedby, “Langevin equations for continuous time Lévy flights,” Physical Review E, vol. 50, no. 2, pp. 1657-1660, 1994.
[204] Y. Fukui and T. Morita, “Derivation of the stationary generalized Langevin equation,” Journal of Physics A, vol. 4, no. 4, pp. 477-490, 1971.
[205] S. C. Kou, “Stochastic modeling in nanoscale biophysics: subdiffusion within proteins,” The Annals of Applied Statistics, vol. 2, no. 2, pp. 501-535, 2008. · Zbl 1400.62272
[206] V. V. Anh, C. C. Heyde, and N. N. Leonenko, “Dynamic models of long-memory processes driven by Lévy noise,” Journal of Applied Probability, vol. 39, no. 4, pp. 730-747, 2002. · Zbl 1016.60039
[207] B. N. N. Achar, J. W. Hanneken, and T. Clarke, “Damping characteristics of a fractional oscillator,” Physica A, vol. 339, no. 3-4, pp. 311-319, 2004.
[208] B. N. N. Achar, J. W. Hanneken, and T. Clarke, “Response characteristics of a fractional oscillator,” Physica A, vol. 309, no. 3-4, pp. 275-288, 2002. · Zbl 0995.70017
[209] C. H. Eab and S. C. Lim, “Fractional generalized Langevin equation approach to single-file diffusion,” Physica A, vol. 389, no. 13, pp. 2510-2521, 2010.
[210] C. H. Eab and S. C. Lim, “Fractional Langevin equations of distributed order,” Physical Review E, vol. 83, no. 3, Article ID 031136, 10 pages, 2011.
[211] S. C. Lim and L. P. Teo, “Modeling single-file diffusion with step fractional Brownian motion and a generalized fractional Langevin equation,” Journal of Statistical Mechanics, vol. 2009, no. 8, Article ID P08015, 2009.
[212] S. C. Lim, L. Ming, and L. P. Teo, “Locally self-similar fractional oscillator processes,” Fluctuation and Noise Letters, vol. 7, no. 2, pp. L169-L179, 2007.
[213] M. Li, S. C. Lim, and S. Chen, “Exact solution of impulse response to a class of fractional oscillators and its stability,” Mathematical Problems in Engineering, vol. 2011, Article ID 657839, 9 pages, 2011. · Zbl 1202.34018
[214] S. C. Lim, C. H. Eab, K. H. Mak, M. Li, and S. Chen, “Solving linear coupled fractional differential equations and their applications,” Mathematical Problems in Engineering, vol. 2012, Article ID 653939, 28 pages, 2012. · Zbl 1264.34009
[215] S. C. Lim, M. Li, and L. P. Teo, “Langevin equation with two fractional orders,” Physics Letters A, vol. 372, no. 42, pp. 6309-6320, 2008. · Zbl 1225.82049
[216] R. A. Gabel and R. A. Roberts, Signals and Linear Systems, John Wiley & Sons, New York, NY, USA, 1973.
[217] M. Carlini, T. Honorati, and S. Castellucci, “Photovoltaic greenhouses: comparison of optical and thermal behaviour for energy savings,” Mathematical Problems in Engineering, vol. 2012, Article ID 743764, 10 pages, 2012. · Zbl 06173540
[218] M. Carlini and S. Castellucci, “Modelling the vertical heat exchanger in thermal basin,” in Proceedings of the ICCSA, 2011, Part 4, vol. 6785 of Springer Lecture Notes in Computer Science, pp. 277-286, Springer, New York, NY, USA, 2011.
[219] M. Carlini, C. Cattani, and A. Tucci, “Optical modelling of square solar con-centrator,” in Proceedins of the ICCSA, 2011, Part 4, vol. 6785 of Springer Lecture Notes in Computer Science, pp. 287-295, Springer, New York, NY, USA, 2011.
[220] L. Qiu, B. G. Xu, and S. B. Li, “H2/H\infty control of networked control system with random time delays,” Science China Information Sciences, vol. 54, no. 12, pp. 2615-2630, 2011. · Zbl 1266.93138
[221] J. Li, J. Z. Wang, S. K. Wang, L. L. Ma, and W. Shen, “Dynamic image stabilization precision test system based on the Hessian matrix,” Science China Information Sciences, vol. 55, no. 9, pp. 2056-2074, 2012. · Zbl 1270.94019
[222] W. X. Zhao and H. F. Chen, “Markov chain approach to identifying Wiener systems,” Science China Information Sciences, vol. 55, no. 5, pp. 1201-1217, 2012. · Zbl 1245.93134
[223] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & sons, NewYork, NY, USA, 1993. · Zbl 0789.26002
[224] T. Hida, Brownian Motion, Springer, New York, NY, USA, 1980. · Zbl 0432.60002
[225] A. H. Zemanian, “An introduction to generalized functions and the generalized Laplace and Legendre transformations,” SIAM Review, vol. 10, no. 1, pp. 1-24, 1968. · Zbl 0177.40905
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.