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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 673648, 23 pages
doi:10.1155/2012/673648
Review Article
On Noise
1Department of Computer and Information Science, University of Macau, Avenida Padre Tomas Pereira, Taipa, Macau
2School of Information Science & Technology, East China Normal University, Shanghai 200062, China
Received 11 October 2012; Accepted 23 October 2012
Academic Editor: Carlo Cattani
Copyright © 2012 Ming Li and Wei Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Linked References
- W. Schottky, “Uber spontane Stromschwankungen in verschiedenen Elektri-zittsleitern,” Annalen der Physik, vol. 362, no. 23, pp. 541–567, 1918. View at Publisher · View at Google Scholar
- W. Schottky, “Zur Berechnung und Beurteilung des Schroteffektes,” Annalen der Physik, vol. 373, no. 10, pp. 157–176, 1922. View at Publisher · View at Google Scholar
- J. B. Johnson, “The Schottky effect in low frequency circuits,” Physical Review, vol. 26, no. 1, pp. 71–85, 1925. View at Publisher · View at Google Scholar · View at Scopus
- B. B. Mandelbrot, Multifractals and 1/f Noise, Springer, New York, NY, USA, 1998. View at Zentralblatt MATH
- 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. View at Publisher · View at Google Scholar · View at Scopus
- E. J. Wagenmakers, S. Farrell, and R. Ratcliff, “Estimation and interpretation of 1/fα noise in human cognition,” Psychonomic Bulletin and Review, vol. 11, no. 4, pp. 579–615, 2004. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar · View at Scopus
- O. Miramontes and P. Rohani, “Estimating 1/fα scaling exponents from short time-series,” Physica D, vol. 166, no. 3-4, pp. 147–154, 2002. View at Publisher · View at Google Scholar · View at Scopus
- C. M. van Vliet, “Random walk and 1/f noise,” Physica A, vol. 303, no. 3-4, pp. 421–426, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- H. J. Jensen, “Lattice gas as a model of 1/f noise,” Physical Review Letters, vol. 64, no. 26, pp. 3103–3106, 1990. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- F. N. Hooge, “1/f noise,” Physica B, vol. 83, no. 1, pp. 14–23, 1976. View at Publisher · View at Google Scholar · View at Scopus
- M. B. Weissman, “Simple model for 1/f noise,” Physical Review Letters, vol. 35, no. 11, pp. 689–692, 1975. View at Publisher · View at Google Scholar · View at Scopus
- F. N. Hooge, “Discussion of recent experiments on 1/f noise,” Physica, vol. 60, no. 1, pp. 130–144, 1972. View at Publisher · View at Google Scholar · View at Scopus
- C. Cattani, “Harmonic wavelet approximation of random, fractal and high frequency signals,” Telecommunication Systems, vol. 43, no. 3-4, pp. 207–217, 2010. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- C. Cattani, E. Laserra, and I. Bochicchio, “Simplicial approach to fractal structures,” Mathematical Problems in Engineering, vol. 2012, Article ID 958101, 21 pages, 2012. View at Publisher · View at Google Scholar
- C. Cattani, “Fractional calculus and Shannon wavelet,” Mathematical Problems in Engineering, vol. 2012, Article ID 502812, 26 pages, 2012. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar
- M. S. Keshner, “1/f noise,” Proceedings of the IEEE, vol. 70, no. 3, pp. 212–218, 1982. View at Publisher · View at Google Scholar · View at Scopus
- B. Ninness, “Estimation of 1/f Noise,” IEEE Transactions on Information Theory, vol. 44, no. 1, pp. 32–46, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- N. J. Kasdin, “Discrete simulation of colored noise and stochastic processes and 1/fα power law noise generation,” Proceedings of the IEEE, vol. 83, no. 5, pp. 802–827, 1995. View at Publisher · View at Google Scholar · View at Scopus
- G. Corsini and R. Saletti, “1/fγ power spectrum noise sequence generator,” IEEE Transactions on Instrumentation and Measurement, vol. 37, no. 4, pp. 615–619, 1988. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Scopus
- 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. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- R. N. Mantegna and H. E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press, Cambridge, UK, 2000.
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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.
- B. J. West and P. Grigolini, “Chipping away at memory,” Biological Cybernetics, vol. 103, no. 2, pp. 167–174, 2010. View at Publisher · View at Google Scholar · View at Scopus
- B. J. West and M. F. Shlesinger, “On the ubiquity of noise,” International Journal of Modern Physics B, vol. 3, no. 6, pp. 795–819, 1989. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- M. Kobayashi and T. Musha, “1/f fluctuation of heartbeat period,” IEEE Transactions on Biomedical Engineering, vol. 29, no. 6, pp. 456–457, 1982. View at Publisher · View at Google Scholar · View at Scopus
- B. Neumcke, “1/f noise in membranes,” Biophysics of Structure and Mechanism, vol. 4, no. 3, pp. 179–199, 1978. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- E. Frehland, “Diffusion as a source of 1/f noise,” The Journal of Membrane Biology, vol. 32, no. 1, pp. 195–196, 1977. View at Publisher · View at Google Scholar · View at Scopus
- M. E. Green, “Diffusion and 1/f noise,” The Journal of Membrane Biology, vol. 28, no. 1, pp. 181–186, 1976. View at Publisher · View at Google Scholar · View at Scopus
- I. Csabai, “1/f noise in computer network traffic,” Journal of Physics A, vol. 27, no. 12, pp. L417–L421, 1994. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- P. Loiseau, P. Gonçalves, 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- A. M. Yaglom, Correlation Theory of Stationary and Related Random Functions, vol. 1, Springer, New York, NY, USA, 1987.
- A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York, NY, USA, 1997.
- B. W. Lindgren and G. W. McElrath, Introduction to Probability and Statistics, The Macmillan, New York, NY, USA, 1959.
- J. L. Doob, “The elementary Gaussian processes,” Annals of Mathematical Statistics, vol. 15, pp. 229–282, 1944. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- A. N. Kolmogorov, Fundamental of Probability, Business Press, Shanghai, China, 1954, Translated from Russian by S.-T. Ding.
- G. E. P. Box, G. M. Jenkins, and G. C. Reinsel, Time Series Analysis: Forecasting and Control, Prentice Hall, Englewood Cliffs, NJ, USA, 1994.
- 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.
- 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.
- J. Beran, Statistics for Long-Memory Processes, vol. 61 of Monographs on Statistics and Applied Probability, Chapman & Hall, New York, NY, USA, 1994.
- J. Beran, “Statistical methods for data with long-range dependence,” Statistical Science, vol. 7, no. 4, pp. 404–416, 1992. View at Publisher · View at Google Scholar
- A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 2, MIT Press, Cambridge, Mass, USA, 1971.
- 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. View at Publisher · View at Google Scholar
- N. C. Nigam, Introduction to Random Vibrations, MIT Press, Cambridge, Mass, USA, 1983.
- T. T. Song and M. Grigoriu, Random Vibration of Mechanical and Structural Systems, Prentice Hall, New York, NY, USA, 1993.
- C. M. Harris, Shock and Vibration Handbook, McGraw-Hill, New York, NY, USA, 4th edition, 1995.
- H. Czichos, T. Saito, and L. Smith, Springer Handbook of Metrology and Testing, Springer, New York, NY, USA, 2011.
- W. N. Sharpe Jr., Springer Handbook of Experimental Solid Mechanics, Springer, New York, NY, USA, 2008.
- C. Tropea, A. L. Yarin, and J. F. Foss, Eds., Springer Handbook of Experimental Fluid Mechanics, Springer, New York, NY, USA, 2007.
- K. H. Grote and E. K. Antonsson, Eds., Springer Handbook of Mechanical Engineering, Springer, New York, NY, USA, 2009.
- R. Kramme, K. P. Hoffmann, and R. S. Pozos, Springer Handbook of Medical Technology, Springer, New York, NY, USA, 2012.
- W. Kresse and D. M. Danko, Springer Handbook of Geographic Information, Springer, New York, NY, USA, 2012.
- S. S. Bhattacharyya, F. Deprettere, R. Leupers, and J. Takala, Eds., Handbook of Signal Processing Systems, Springer, New York, NY, USA, 2010.
- S. K. Mitra and J. F. Kaiser, Handbook for Digital Signal Processing, John Wiley & Sons, New York, NY, USA, 1993.
- W. A. Woyczyński, A First Course in Statistics for Signal Analysis, Birkhäuser, Boston, Mass, USA, 2006.
- R. A. Bailey, Design of Comparative Experiments, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, UK, 2008. View at Publisher · View at Google Scholar
- ASME, Measurement Uncertainty Part 1, Instruments and Apparatus, Sup-plement to ASME, Performance Test Codes, ASME, New York, NY, USA, 1986.
- D. Sheskin, Statistical Tests and Experimental Design: A Guidebook, Gardner Press, New York, NY, USA, 1984.
- T. W. MacFarland, Two-Way Analysis of Variance, Springer, New York, NY, USA, 2012.
- 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. View at Publisher · View at Google Scholar
- P. Fieguth, Statistical Image Processing and Multidimensional Modeling, Information Science and Statistics, Springer, New York, NY, USA, 2011. View at Publisher · View at Google Scholar
- J. Nauta, Statistics in Clinical Vaccine Trials, Springer, New York, NY, USA, 2011.
- A. Gelman, “Analysis of variance—why it is more important than ever,” The Annals of Statistics, vol. 33, no. 1, pp. 1–53, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- C. G. Pendse, “A note on mathematical expectation,” The Mathematical Gazette, vol. 22, no. 251, pp. 399–402, 1938.
- B. B. Mandelbrot, Gaussian Self-Affinity and Fractals, Springer, New York, NY, USA, 2001.
- G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Stochastic Modeling, Chapman & Hall, New York, NY, USA, 1994.
- 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. View at Publisher · View at Google Scholar · View at Scopus
- I. M. Gelfand and K. Vilenkin, Generalized Functions, vol. 1, Academic Press, New York, NY, USA, 1964.
- M. Li and S. C. Lim, “Power spectrum of generalized Cauchy process,” Telecommunication Systems, vol. 43, no. 3-4, pp. 219–222, 2010. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar
- B. B. Mandelbrot and J. W. van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Review, vol. 10, pp. 422–437, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- P. Flandrin, “On the spectrum of fractional Brownian motions,” IEEE Transactions on Information Theory, vol. 35, no. 1, pp. 197–199, 1989. View at Publisher · View at Google Scholar
- M. Li, “Fractal time series—a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- 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. View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- 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. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- M. Li and S. C. Lim, “Modeling network traffic using generalized Cauchy process,” Physica A, vol. 387, no. 11, pp. 2584–2594, 2008. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- W. A. Woodward, Q. C. Cheng, and H. L. Gray, “A -factor GARMA long-memory model,” Journal of Time Series Analysis, vol. 19, no. 4, pp. 485–504, 1998. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar
- C. Weng and Y. Zhang, “Characterization of multivariate heavy-tailed distribution families via copula,” Journal of Multivariate Analysis, vol. 106, pp. 178–186, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- P. Embrechts, J. Nešlehová, 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- R. Ibragimov, “Heavy-tailedness and threshold sex determination,” Statistics and Probability Letters, vol. 78, no. 16, pp. 2804–2810, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- S. Resnick, “On the foundations of multivariate heavy-tail analysis,” Journal of Applied Probability, vol. 41, pp. 191–212, 2004. View at Zentralblatt MATH · View at Scopus
- 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. View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- V. Limic, “A LIFO queue in heavy traffic,” The Annals of Applied Probability, vol. 11, no. 2, pp. 301–331, 2001. View at Zentralblatt MATH · View at Scopus
- H. Le and A. O. 'Hagan, “A class of bivariate heavy-tailed distributions,” San-Khyā: The Indian Journal of Statistics, Series B, vol. 60, no. 1, pp. 82–100, 1998. View at Zentralblatt MATH
- M. C. Bryson, “Heavy-tailed distributions: properties and tests,” Technometrics, vol. 16, no. 1, pp. 61–68, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- J. Beran, “Discussion: heavy tail modeling and teletraffic data,” The Annals of Statistics, vol. 25, no. 5, pp. 1852–1856, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar
- V. Pisarenko and M. Rodkin, Heavy-Tailed Distributions in Disaster Analysis, Springer, New York, NY, USA, 2010.
- S. I. Resnick, Heavy-Tail Phenomena Probabilistic and Statistical Modeling, Springer, New York, NY, USA, 2007, Probabilistic and statistical modeling.
- 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.
- 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. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- H. E. Stanley, “Power laws and universality,” Nature, vol. 378, no. 6557, p. 554, 1995. View at Scopus
- I. Eliazar and J. Klafter, “A probabilistic walk up power laws,” Physics Reports, vol. 511, no. 3, pp. 143–175, 2012. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar · View at Scopus
- A. Saiz, “Boltzmann power laws,” Physica A, vol. 389, no. 2, pp. 225–236, 2010. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- T. Simon, “Fonctions de Mittag-Leffler et processus de Lévy stables sans sauts négatifs,” Expositiones Mathematicae, vol. 28, no. 3, pp. 290–298, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- R. Lambiotte and L. Brenig, “Truncated Lévy distributions in an inelastic gas,” Physics Letters A, vol. 345, no. 4–6, pp. 309–313, 2005. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- P. Garbaczewski, “Cauchy flights in confining potentials,” Physica A, vol. 389, no. 5, pp. 936–944, 2010. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar
- G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill, New York, NY, USA, 1961.
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- I. A. Lubashevsky, “Truncated Lévy flights and generalized Cauchy processes,” European Physical Journal B, vol. 82, no. 2, pp. 189–195, 2011. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar · View at Scopus
- E. E. Kuruoǧlu, “Density parameter estimation of skewed α-stable distributions,” IEEE Transactions on Signal Processing, vol. 49, no. 10, pp. 2192–2201, 2001. View at Publisher · View at Google Scholar · View at Scopus
- A. P. Petropulu, J. C. Pesquet, X. Yang, and J. J. Yin, “Power-law shot noise and its relationship to long-memory α-stable processes,” IEEE Transactions on Signal Processing, vol. 48, no. 7, pp. 1883–1892, 2000. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- L. Landau, “On the energy loss of fast particles by ionization,” Journal of Physics, vol. 8, pp. 201–205, 1944.
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- J. Holtsmark, “Uber die Verbreiterung von Spektrallinien,” Annalen der Physik, vol. 363, no. 7, pp. 577–630, 1919. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- 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. View at Publisher · View at Google Scholar · View at Scopus
- R. G. Garroppo, S. Giordano, M. Pagano, and G. Procissi, “Testing α-stable processes in capturing the queuing behavior of broadband teletraffic,” Signal Processing, vol. 82, no. 12, pp. 1861–1872, 2002. View at Publisher · View at Google Scholar · View at Scopus
- J. R. Gallardo, D. Makrakis, and L. Orozco-Barbosa, “Use of α-stable self-similar stochastic processes for modeling traffic in broadband networks,” Performance Evaluation, vol. 40, no. 1, pp. 71–98, 2000. View at Publisher · View at Google Scholar · View at Scopus
- A. Karasaridis and D. Hatzinakos, “Network heavy traffic modeling using α-stable self-similar processes,” IEEE Transactions on Communications, vol. 49, no. 7, pp. 1203–1214, 2001. View at Publisher · View at Google Scholar · View at Scopus
- P. R. de Montmort, “Essay d'analyse sur les jeux de hazard,” 1713.
- P. R. de Montmort, “Essay d'analyse sur les jeux de hazard,” American Mathematical Society, 1980.
- D. Bernoulli, “Exposi-tion of a new theory on the measurement of risk,” Econometrica, vol. 22, no. 1, pp. 22–36, 1954.
- W. T. Coffey, Y. P. Kalmykov, and J. T. Waldron, The Langevin Equation, World Scientific, Singapore, 2nd edition, 2004.
- 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. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar
- D. Panja, “Generalized langevin equation formulation for anomalous polymer dynamics,” Journal of Statistical Mechanics, vol. 2010, no. 2, Article ID L02001, 2010. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- E. Lutz, “Fractional Langevin equation,” Physical Review E, vol. 64, no. 5, Article ID 051106, 4 pages, 2001. View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- K. S. Fa, “Fractional Langevin equation and Riemann-Liouville fractional derivative,” European Physical Journal E, vol. 24, no. 2, pp. 139–143, 2007. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- 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. View at Publisher · View at Google Scholar · View at Scopus
- H. C. Fogedby, “Langevin equations for continuous time Lévy flights,” Physical Review E, vol. 50, no. 2, pp. 1657–1660, 1994. View at Publisher · View at Google Scholar · View at Scopus
- Y. Fukui and T. Morita, “Derivation of the stationary generalized Langevin equation,” Journal of Physics A, vol. 4, no. 4, pp. 477–490, 1971. View at Publisher · View at Google Scholar · View at Scopus
- S. C. Kou, “Stochastic modeling in nanoscale biophysics: subdiffusion within proteins,” The Annals of Applied Statistics, vol. 2, no. 2, pp. 501–535, 2008. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- 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. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
- R. A. Gabel and R. A. Roberts, Signals and Linear Systems, John Wiley & Sons, New York, NY, USA, 1973.
- 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. View at Publisher · View at Google Scholar
- 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.
- 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.
- L. Qiu, B. G. Xu, and S. B. Li, “H2/ control of networked control system with random time delays,” Science China Information Sciences, vol. 54, no. 12, pp. 2615–2630, 2011. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & sons, NewYork, NY, USA, 1993.
- T. Hida, Brownian Motion, Springer, New York, NY, USA, 1980.
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH