Abstract
Stable distributions are special types of probability distributions whose origin is a particular limit regime of other types of distributions. They are closely related to the simple convolution process, which is introduced first for continuous and then for discrete random variables. This leads to the central limit theorem as one of the most important results of probability theory, as well as to its generalized version which is useful in the analysis of random walks. Extreme-value distributions are also presented, as they possess a limit theorem of their own (Fisher–Tippett–Gnedenko). The last part is devoted to the discussion of discrete-time and continuous-time random walks, together with their characteristic diffusion properties.
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References
D.L. Evans, L.M. Leemis, Algorithms for computing the distributions of sums of discrete random variables. Math. Comp. Model. 40, 1429 (2004)
W. Feller, An Introduction to Probability Theory and Its Applications, vol. 2, 2nd edn. (Wiley, New York, 1971)
I.S. Tyurin, On the accuracy of the Gaussian approximation. Dokl. Math. 80, 840 (2009)
A. Lyon, Why are normal distributions normal? Brit. J. Phil. Sci. 65, 621 (2014)
I. Kuščer, A. Kodre, Mathematik in Physik und Technik (Springer, Berlin, 1993)
J.P. Nolan, Stable Distributions—Models for Heavy Tailed Data (Birkhäuser, Boston, 2010)
S. Borak, W. Härdle, R. Weron, Stable Distributions, SFB 649 Discussion Paper 2005–008 (Humboldt University Berlin, Berlin, 2005)
S. Širca, M. Horvat, Computational Methods for Physicists (Springer, Berlin, 2012)
GSL (GNU Scientific Library), http://www.gnu.org/software/gsl
G.G. Márquez, One Hundred Years of Solitude (HarperCollins, New York, 2006)
E.J. Gumbel, Statistics of Extremes (Columbia University Press, New York, 1958)
S. Coles, An Introduction to Statistical Modeling of Extreme Values (Springer, Berlin, 2001)
M.R. Leadbetter, G. Lindgren, H. Rootzén, Extremes and Related Properties of Random Sequences and Processes (Springer, New York, 1983)
S.I. Resnick, Extreme Values, Regular Variation, and Point Processes (Springer, New York, 1987)
R.A. Fisher, L.H.C. Tippett, On the estimation of the frequency distribution of the largest or smallest member of a sample. Proc. Camb. Phil. Soc. 24, 180 (1928)
B.V. Gnedenko, Sur la distribution limite du terme maximum d’une série aléatoire. Ann. Math. 44, 423 (1943)
MeteoSwiss IDAWEB, http://gate.meteoswiss.ch/idaweb/
B.D. Hughes, Random Walks and Random Environments: Vol. 1: Random Walks (Oxford University Press, New York, 1995)
M. Bazant, Random walks and diffusion, MIT OpenCourseWare, Course 18.366, http://ocw.mit.edu/courses/mathematics/
E.W. Montroll, G.H. Weiss, Random walks on lattices II. J. Math. Phys. 6, 167 (1965)
R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1 (2000)
M.F. Shlesinger, B.J. West, J. Klafter, Lévy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58, 1100 (1987)
V. Tejedor, R. Metzler, Anomalous diffusion in correlated continuous time random walks. J. Phys. A: Math. Theor. 43, 082002 (2010)
J.J. Brehm, W.J. Mullin, Introduction to the Structure of Matter (Wiley, New York, 1989)
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Širca, S. (2016). Stable Distributions and Random Walks. In: Probability for Physicists. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-31611-6_6
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DOI: https://doi.org/10.1007/978-3-319-31611-6_6
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