A probabilistic walk up power laws

Physics Reports

Volumn 511, Issue 3, 2012, Pages 143-175

  Eliazar, Iddo ^a   Klafter, Joseph ^b  

^a HOLON INSTITUTE OF TECHNOLOGY   (Israel)

^b TEL AVIV UNIVERSITY   (Israel)

Author keywords

1 f noises; Anomalous diffusion; Anomalous relaxation; Central Limit Theorems; Exceedances; Joseph effect; L vy laws; Long range dependence; Lorenz curve; Noah effect; Pareto's law; Pareto's Poisson process; Poisson processes; Power law statistics; Rank distributions; Sub diffusion; Super diffusion; Universality

Indexed keywords

EID: 84856090260     PISSN: 03701573     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physrep.2011.11.001     Document Type: Review

References (143)

1. http://www.nslij-genetics.org/wli/zipf/.
2. V. Pareto, Cours d'économie politique (Droz, Geneva, 1896).
3. F. Auerbach, Petermanns Geographische Mitteilungen, 59 (1913) 74.
4. J.B. Estoup, Gammes stenographiques, Institut Stenographique de France, Paris, 1916.
5. Lotka A.J. J. Washington Acad. Sci. 1926, 16:317.
6. Mandelbrot B.B. Fractals and Scaling in Finance: Discontinuity, Concentration, Risk 1997, Springer, New York.
7. Gabaix X. Ann. Rev. Econ. 2009, 1:255.
8. Albert R., Barabási A.L. Rev. Modern Phys. 2002, 74:47.
9. Cohen R., Havlin S. Complex Networks: Structure, Robustness and Function 2010, Cambridge University Press, Cambridge.
10. Mitzenmacher M. Internet Math. 2004, 1:226.
11. Newman M.E.J. Contemp. Phys. 2005, 46:323.
12. Clauset A., Shalizi C.R., Newman M.E.J. SIAM Rev. 2009, 51:661.
13. Klafter J., Sokolov I.M. First Steps in Random Walks 2011, Oxford University Press, Oxford.
14. Einstein A. Ann. Phys. 1905, 17:549.
15. von Smoluchowski M. Ann. Phys. 1906, 21:756.
16. Metzler R., Klafter J. Phys. Rep. 2000, 339:1.
17. Klafter J., Sokolov I.M. Phys. World 2005, 18:29.
18. Shlesinger M.F., Zaslavsky G.M., Klafter J. Nature 1993, 363:31.
19. Shlesinger M.F. Nature 2001, 411:641.
20. Mandelbrot B.B., Wallis J.R. Water Resour. Res. 1968, 4:909.
21. Cox D.R. Long-range dependence: A review. Statistics: An Appraisal 1984, 55-74. Iowa State University Press, Ames. H.A. David, H.T. David (Eds.).
22. Processes with long-range Correlations: Theory and Applications 2003, Springer-Verlag, New York. G. Rangarajan, M. Ding (Eds.).
23. Kesner M.S. Proc. IEEE 1982, 70:212.
24. van der Ziel A. Proc. IEEE 1988, 76:233.
25. Shlesinger M.F., Montroll E.W. Proc. Natl. Acad. Sci. USA 1984, 81:1280.
26. Klafter J., Shlesinger M.F. Proc. Natl. Acad. Sci. USA 1986, 83:848.
27. Malkiel B.G. A Random Walk Down Wall Street 2011, Norton & Co, New York.
28. Zipf G.K. Human Behavior and the Principle of Least Effort 1949, Addison-Wesley, Cambridge.
29. Lorenz M.O. Pub. Am. Stat. Assoc. 1905, 9:209.
30. Pietra G. Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti 1914-15, LXXIV, parte II:775.
31. Bingham N.H., Goldie C.M., Teugels J.L. Regular Variation 1987, Cambridge University Press, Cambridge.
32. Mishima O., Stanley H.E. Nature 1998, 396:329.
33. Stanley H.E. Introduction to Phase Transitions and Critical Phenomena 1987, Oxford University Press, Oxford.
34. Feller W. An Introduction to Probability Theory and its Applications, vol. 2 1971, Wiley, New York. 2nd ed.
35. A Practical Guide to Heavy Tails: Statistical Techniques and Applications 1998, Birkhauser, Boston. R.J. Adler, R.E. Feldman, M.S. Taqqu (Eds.).
36. Mantegna R.N., Stanley H.E. Introduction to Econophysics: Correlations and Complexity in Finance 2007, Cambridge University Press, Cambridge.
37. Eliazar I. J. Phys. A: Math. Theor. 2011, 44:022001.
38. Eliazar I., Cohen M.H. Physica A 2011, 390:4293.
39. Eliazar I. Physica A 2007, 386:318.
40. Coulter P.B. Measuring Inequality: A Methodological Handbook 1989, Westview Press, Boulder.
41. Cowell F.A. Measuring Inequality 2000, http://sticerd.lse.ac.uk/research/frankweb/MeasuringInequality/index.html, 3rd ed.
42. I. Eliazar, I.M. Sokolov, Physica A ("Measuring statistical evenness: A panoramic overview") (2011), in press (). doi:10.1016/j.physa.2011.09.007.
43. Gumbel E.J. Statistics of Extremes 1958, Columbia University Press, New York.
44. Galambos J. Asymptotic Theory of Extreme Order Statistics 1987, Krieger, New York. 2nd ed.
45. Lévy P. Théorie de l'addition des variables Aléatoires 1954, Gauthier-Villars, Paris.
46. Gnedenko B.V., Kolmogorov A.N. Limit Distributions for Sums of Independent Random Variables 1954, Addison-Wesley, London.
47. Kingman J.F.C. Poisson Processes 1993, Oxford University Press, Oxford.
48. Eliazar I., Klafter J. J. Stat. Phys. 2008, 131:487.
49. Eliazar I., Klafter J. Europhys. Lett. 2008, 83:40004.
50. Hill B.M. Ann. Stat. 1975, 3:1163.
51. Eliazar I., Klafter J. Phys. Rev. E 2009, 79:011103.
52. Eliazar I. Prob. Eng. Inform. Sci. 2005, 19:289.
53. Eliazar I., Sokolov I.M. Physica A 2010, 389:4462.
54. Eliazar I., Klafter J. Physica A 2008, 387:4985.
55. Eliazar I., Klafter J. Chem. Phys. 2010, 370:290.
56. Eliazar I., Klafter J. Phys. Rev. E 2010, 82:021122.
57. Eliazar I., Klafter J. Phys. Rev. E 2008, 77:061125.
58. Eliazar I., Klafter J. Risk Decis. Anal. 2009, 1:155.
59. Eliazar I., Klafter J. Physica A 2007, 383:171.
60. Eliazar I., Sokolov I.M. Phys. Rev. E 2010, 81:011122.
61. Pearson K. Nature 1905, 72:294.
62. Weiss G.H. Aspects and Applications of the Random Walk 1994, North-Holland, Amsterdam.
63. Rice S.O. Bell Syst. Tech. J. 1944, 23:282.
64. Rice S.O. Bell Syst. Tech. J. 1945, 24:4.
65. Lowen S.B., Teich M.C. Phys. Rev. Lett. 1989, 63:1755.
66. Lowen S.B., Teich M.C. IEEE Trans. Inform. Theor. 1990, 36:1302.
67. Gross D., Harris C.M. Fundamentals of Queueing Theory 1974, Wiley, New York.
68. Takacs L. Introduction to the Theory of Queues 1962, Oxford University Press, Oxford.
69. Eliazar I. Queueing Syst. 2007, 55:71.
70. Van Kampen N.G. Stochastic Processes in Physics and Chemistry 2007, Elsevier, Amsterdam. 3rd ed.
71. Bouchaud J.P., Georges A. Phys. Rep. 1990, 195:12.
72. Scher H., Montroll E.W. Phys. Rev. B 1975, 12:2455.
73. Kirchner J.W., Feng X., Neal C. Nature 2000, 403:524.
74. Golding I., Cox E.C. Phys. Rev. Lett. 2006, 96:098102.
75. Szymanski J., Weiss M. Phys. Rev. Lett. 2009, 103:038102.
76. Richardson L.F. Proc. Roy. Soc. 1926, 110:709.
77. Solomon T., Weeks E., Swinney H. Phys. Rev. Lett. 1993, 71:3975.
78. Caspi A., Granek R., Elbaum M. Phys. Rev. Lett. 2000, 85:5655.
79. Sims D.W., et al. Nature 2008, 451:1098.
80. Liu B., Goree J. Phys. Rev. Lett. 2008, 100:055003.
81. Coffey W.T., Kalmykov Yu.P., Waldron J.T. The Langevin Equation 2004, World Scientific, Singapore. 2nd ed.
82. Condamin S., et al. Nature 2007, 450:77.
83. Brockmann D., Hufnagel L., Geisel T. Nature 2006, 439:462.
84. Gonzalez M.C., Hidalgo C.A., Barabasi A.L. Nature 2008, 453:779.
85. Barthelemy P., Bertolotti J., Wiersma D.S. Nature 2008, 453:495.
86. Chechkin A.V., Gonchar V.Y., Szydlowski M. Phys. Plasmas 2002, 9:78.
87. Watkins N., et al. Space Sci. Rev. 2005, 121:271.
88. Weron A., et al. Phys. Rev. E 2005, 71:016113.
89. Benson D.A., Wheatcraft S.W., Meerschaert M.M. Water Resour. Res. 2000, 36:1403.
90. Benson D.A., Schumer R., Meerschaert M.M., Wheatcraft S.W. Transp. Porous Media 2001, 42:211.
91. Deng Z.Q., Bengtsson L., Singh V.P. Environ. Fluid Mech. 2006, 6:451.
92. Kim S., Kavvas M.L. J. Hydrol. Eng. 2006, 11:80.
93. Shen C., Phanikumar M.S. Adv. Water Resour. 2009, 32:1482.
94. Ganti V., Meerschaert M.M., Foufoula Georgiou E., Viparelli E., Parker G. J. Geophys. Res. 2010, 115:F00A12.
95. Kelly J.F., McGough R.J., Meerschaert M.M. J. Acoust. Soc. Am. 2008, 124:2861.
96. Carasso A.S. Opt. Eng. 2006, 4510:107004.
97. http://www.nslij-genetics.org/wli/1fnoise.
98. Mandelbrot B.B., Van Ness J.W. SIAM Rev. 1968, 10:422.
99. Mandelbrot B.B. Multifractals and 1/f Noise 1999, Springer, New York.
100. Peng C.K., et al. Nature 1992, 356:168.
101. Buldyrev S.V., et al. Phys. Rev. E 1995, 51:5084.
102. Hausdorff J.M., et al. J. Appl. Physiol. 1995, 78:349.
103. Segre P.N., Herbolzheimer E., Chaikin P.M. Phys. Rev. Lett. 1997, 79:2574.
104. Koscielny-Bunde E., et al. Phys. Rev. Lett. 1998, 81:729.
105. Parameswaran Gopikrishnan Y.L., et al. Phys. Rev. E 1999, 60:1390.
106. Ivanov P.C., et al. Chaos 2001, 11:641.
107. Campillo M., Paul A. Science 2003, 299:547.
108. Coffey W.T., Kalmykov Yu.P. Fractals, Diffusions and Relaxation in Disordered Systems 2006, Wiley-Interscience, New York.
109. Williams G., Watts D.C. Trans. Faraday Soc. 1970, 66:80.
110. Kriza G., Mihaly G. Phys. Rev. Lett. 1986, 56:2529.
111. Degiorgio V., et al. Phys. Rev. Lett. 1990, 64:1043.
112. Dixon P.K., Wu L., Nagel S.R. Phys. Rev. Lett. 1990, 65:1108.
113. Phillips J.C. Rep. Prog. Phys. 1996, 59:1133.
114. Rabani E., Gezelter J.D., Berne B.J. Phys. Rev. Lett. 1999, 82:3649.
115. Redner S. A Guide to First-passage Processes 2001, Cambridge University Press, Cambridge.
116. Eliazar I., Klafter J. J. Phys. A 2009, 42:472003.
117. Eliazar I., Klafter J. J. Phys. A 2010, 43:132002.
118. Eliazar I., Klafter J. Proc. Nat. Acad. Sci. (USA) 2009, 106:12251.
119. Eliazar I., Klafter J. Phys. Rev. E 2010, 82:021109.
120. Eliazar I., Klafter J. Phys. Rev. E 2010, 82:011112.
121. Sancho J.M., Lacasta A.M., Lindenberg K., Sokolov I.M., Romero A.H. Phys. Rev. Lett.. Phys. Rev. Lett. 2005, 94:188902.
122. Khoury M., Lacasta A.M., Sancho J.M., Lindenberg K. Phys. Rev. Lett. 2011, 106:090602.
123. Eliazar I., Klafter J. Ann. Phys. 2011, 326:2517.
124. Eliazar I., Klafter J. Phys. Rev. Lett. 2009, 103:040602.
125. Eliazar I., Klafter J. J. Phys. A 2011, 44:222001.
126. Eliazar I., Klafter J. J. Phys. A 2011, 44:405006.
127. Eliazar I., Klafter J. J. Phys. A 2011, 44:415003.
128. I. Eliazar, M.F. Shlesinger, J. Stat. Phys. ("Stochastic flow cascades") (2011), in press (). doi:10.1007/s10955-011-0340-6.
129. Lowen S.B., Teich M.C. Fractal-based point Processes 2005, Wiley, New York.
130. Eliazar I., Klafter J. J. Phys. A 2007, 40:F307.
131. Eliazar I., Klafter J. Physica A 2007, 376:1.
132. Magdziarz M. Stochastic Process. Appl. 2009, 119:3416.
133. Weron A., Magdziarz M. Phys. Rev. Lett. 2010, 105:260603.
134. Magdziarz M., Weron A. Ann. Phys. 2011, 326:2431.
135. Donsker M.D. Mem. Amer. Math. Soc. 1951, 6:1.
136. Brown R. Phil. Mag. 1828, 4:161.
137. Bachelier L. Ann. Sci. Ec. Norm. Super. 1900, 17:21.
138. Bachelier L. Louis Bachelier's Theory of Speculation: The Origins of Modern Finance (M. Davis and A. Etheridge, Trans.) 2006, Princeton University Press, Princeton.
139. Wiener N. J. Math. Phys. 1923, 2:131.
140. Itō K. Mem. Amer. Math. Soc. 1951, 4:1.
141. Janicki A., Weron A. Simulation and Chaotic Behavior of Stable Stochastic Processes 1994, Marcel Dekker, New York.
142. Samrodintsky G., Taqqu M.S. Stable non-Gaussian Random Processes 1994, Chapman and Hall, New York.
143. Eliazar I., Klafter J. J. Stat. Phys. 2005, 119:165.