Mathematical Methods and Concepts for the Analysis of Extreme Events

Frontiers Collection

Volumn Part F934, Issue , 2006, Pages 47-68

  Albeverio, Sergio ^a   Piterbarg, Vladimir ^b  

^a UNIVERSITY OF BONN   (Germany)

^b MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

Catastrophe Theory; Extremal Index; Fractional Brownian Motion; Gaussian Process; Point Process

Indexed keywords

EID: 85166098751     PISSN: 16123018     EISSN: 21976619     Source Type: Book Series    
DOI: 10.1007/3-540-28611-X_3     Document Type: Chapter

References (74)

1. Gumbel E.J., Statistics of extremes. Columbia University Press, New York (1958)
2. Lieblein J., Two early papers on the relation between extreme valued and tensile strength. Biometrika, 41, 559–560 (1954)
3. David H.A., Order statistics. Wiley, New York (1981)
4. Galambos J., The asymptotic theory of extreme order statistics, 2nd edn. Krieger, Malabar, FL (1987)
5. Pfeifer D., Einführung in die Extremwertstatistik. Teubner, Stuttgart (1989)
6. Gottschalk L., Olivry J.C., Reed D., Rosbjerg D., eds, Hydrological extremes: understanding, predicting, mitigating. IAHS, Wallingford, UK, Publication 255 (1999)
7. Kallenberg O., Random measures. Akademie-Verlag, Berlin (1983)
8. Leadbetter M.R., Lindgren G., Rootzén H., Extremes and related properties of random sequences and process. Springer, Berlin Heidelberg New York (1983)
9. Resnick S.I., Extreme values, regular variation, and point processes. Springer, Berlin Heidelberg New York (1987)
10. Embrechts P., Klüppelberg C., Mikosch T., Modelling extremal events for insurance and finance. Springer, Berlin Heidelberg New York (1997)
11. Falk M., Hüsler J., Reiss R.D., Laws of small numbers: extremes and rare events. Birkhäuser, Basel (1994)
12. Piterbarg V.I., Asymptotic methods in the theory of Gaussian processes and fields. Translation Monographs AMS Series, 148 (1996) (processed edition; original in Russian, 1988, Moscow UP)
13. Aldous D., Probability approximations via the Poisson clumping heuristic. Springer, Berlin Heidelberg New York (1989)
14. Ibragimov I.A., Rosanov Yu.A., Gaussian random processes. Springer, Berlin Heidelberg New York (1978)
15. Coles S., An introduction to statistical modeling of extreme values. Springer, Berlin Heidelberg New York (2001)
16. Beirlant J., Goegebeur Y., Segers J., Teugels J., Statistics of extremes: theory and applications. Wiley, Chichester, UK (2004)
17. Kotz S., Nadarajah S., Extreme value distributions: theory and applications. Imperial College Press, London (2000)
18. Castillo E., Extreme value theory in engineering (Statistical Modeling and Decision Science Series). Academic, New York (1988)
19. Benz W., Gumbel E.J., Die Karriere eines deutschen Pazifisten. In: Walberer U. (Hrsg), 10. Mai 1933: Bücherverbrennung in Deutschland und die Folgen. Fischer Taschenbuch, Frankfurt (1983)
20. Deuschel J.D., Stroock D.W., Large deviations. Academic, New York (1989)
21. Harter H.L., A bibliography of extreme value theory. lnt Stat Rev, 46, 279– 306 (1978)
22. Berman S.M., Sojourns and extremes of stochastic processes. Wadsworth, Belmont, CA (1992)
23. Reiss R.-D., Thomas M., Statistical analysis of extreme values: with applications to insurance, finance, hydrology, and other fields, extended 2nd edn. Birkhäuser, Basel (2001)
24. Einmahl H.J., de Haan L., Empirical processes and statistics of extreme values, 1 and 2. AIO Course, available at www.few.eur.nl/few/people/ ldehaan/aio/aio1.ps, www.few.eur.nl/few/people/ldehaan/aio/aio2.ps
25. Freidlin M., Wentzell A., Random perturbations of dynamical systems. Springer, Berlin Heidelberg New York (1984)
26. Dembo A., Zietouni O., Large deviation techniques and applications, Springer, Berlin Heidelberg New York (1998)
27. Cramer H., Leadbetter M.R., Stationary and related stochastic processes. Wiley, Chichester, UK (1967)
28. de Haan L., A spectral representation for max-stable processes. Ann Probab 12, 1194–1204 (1984)
29. Samorodnitsky G., Taqqu M., Stable non-Gaussian random processes. Chapman & Hall, New York (1994)
30. Piterbarg V.I., Fatalov V.R., The Laplace method for probability measures in Banach spaces. Uspekhi Mat Nauk, 50, 9, 57–150 (1995) (English translation: Russian Math Surveys 50, 1151–1239 (1995))
31. Kinnison R.R., Applied extreme value statistics. Battelle, Columbus, OH (1985)
32. Weibull W., A statistical theory of the strength of materials. Ing Akad Handl, 151 (1939)
33. Scafetta N., West B.J., Multiscaling corporative analysis of time series and a discussion on earthquake conversations. Phys Rev Lett 92, 138501–138504 (2004)
34. Hergarten S., Self-organized criticality in earth systems. Springer, Berlin Heidelberg New York (2002)
35. Albeverio S., Shelkovich V., Delta shock waves in multidimensional nonconservative systems of zero pressure gas dynamics. Bonn Preprint 176 (2004)
36. Albeverio S., Ferrario B., Uniqueness of solutions of the stochastic Navier Stokes equation with invariant measure given by the enstrophy. Ann Prob S2, 1632–1649 (2004)
37. Gomez M.E., Dynamic probabilistic models and social structure. Kluwer, Dordrecht (1992)
38. Albeverio S., Hoegh-Krohn R., Mazzucchi S., Mathematical theory of Feynman path integrals – An introduction, 2nd edn. Springer, Berlin Heidelberg New York (2005)
39. Simon B., Functional integration and quantum physics. Academic, New York (1979)
40. Kleinert H., Path integrals in quantum mechanics, statistics, polymer physics and finance. World Scientific, Singapore (2004)
41. Arnold V.I., Catastrophe theory. Springer, Berlin (1984)
42. Albeverio S., Liang S., Asymptotic expansions for the Laplace approximation of sums of Banach space-valued random variables. Ann Prob, 33, 1, 300–336 (2005)
43. Albeverio S., Röckner H., Steblovskaya V., Asymptotic expansions for Ornstein-Uhlenbeck semigroups perturbed by potential over Banach spaces. Stoch Repts, 69, 41–100 (2000)
44. Tiago de Oliveira J., Statistical experiments and applications. D. Ruidel, Docheckz (1984)
45. Beinlart J., Tengeb J.L., Vynckier P., Practical analysis of extreme values. Leuven University Press, Belgium (1996)
46. Chang I, Stauffer D., Pandey R.B., Asymmetries, correlations and fat tails in percolation market model. eprint arXiv cond-mat/0108345V1 (2001)
47. Matthews R., Far out forecasting. New Scientist, 37–40 (1996)
48. Smith R., Extreme value theory. In: Handbook of applicable mathematics, Supplement. Wiley, Chichester, UK, 437–439 (1990)
49. De Haan L., On regular variation and its application to the weak convergence of sample extremes. Math Center Facts, Amsterdam (1971)
50. Coles S.G., Tawn J.A., Modelling extreme multivariate events. J R Stat Soc B, 53, 377–392 (1991)
51. Badii R., Politi A., Complexity: Hierarchical structures and scaling in physics. Cambridge Univ Press, Cambridge (1997)
52. Sornette D., Critical phenomena in natural sciences. Chaos, fractal, selforganization and disorder: concepts and tools. Springer, Berlin Heidelberg New York (2000)
53. Barnett V., The ordering of multivariate data. J R Stat Soc A, 139, 318–345 (1976)
54. Christopeit N., Estimating parameters of an extreme value distribution by the method of moments. J Nat Plann Infer 41, 173–186 (1994)
55. Cessac B., Blanchard Ph., Krüger T., Meunier J.L., Self-organized criticality, thermodynamic formalism. J Stat Phys, 11, 1283–1326 (2004)
56. Ruelle D., Chance and chaos. Princeton University Press, New York (1991)
57. Berger A., Chaos and chance. Walter de Gruter, Berlin (2001)
58. Jain S., Krishna S., Graph theory and the evolution of antocatalytic networks. In: Handbook of graphs and networks. Wiley-VCH, Weinheim, 355– 395 (2003)
59. Gabaix X., Gopikrishnan P., Pleron V., Stanley H.E., A theory of power-law distribution in financial market situation. Nature, 423, 267–276 (2003)
60. Arneido A., Muzy J.F., Sornette D., Direct causal cascade in the stock market. Eur Phy J B 2, 277–282 (1998)
61. Blanchard Ph., Hongler M.O., How many blocks can children pile up – some analytical results. J Phys Soc Jap, 71, 9–11 (2002)
62. Bak P., How nature works. Springer, Berlin Heidelberg New York (1998)
63. Jensen H.J., Self-organized criticality: emerging complex behaviour in physical and biological systems. Cambridge Univ Press, Cambridge (1998)
64. Hahn M.G., Mason D.M., Wiener D.C., eds, Sums, trimmed sums and extremes. Birkhäuser, Boston, MA (1991)
65. Sornette D., Why stock markets crash. Critical events in complex financial systems. Princeton Univ Press, New York (2003)
66. Mandelbrot B.B., Fractals and scaling in finance: discontinuity, concentration, risk. Springer, Berlin Heidelberg New York (1997)
67. Balakrishnan V., Nicolis C., Nicolis G., Extreme value distributions in chaotic dynamics. J Stat Phys, 80, 307–336 (1995)
68. Lamper D., Howison S.D., Johnson N.F., Predictability of large future changes in a competitive evolving population. Phys Rev Lett, 88, 017902-1-4 (2002)
69. Einmahl J., de Haan L., Piterbarg V.I., Non-parametric estimation of the spectral measure of an extreme value distribution. Ann Stat, 29, 1401–1423 (2001)
70. Gomes M.I., Martins M.J., Neves M., Generalized jacknife semi-parametric estimation of the tail index. Pas Math, 59, 393–406 (2002)
71. Kunz A., Personal webpage. See http://www-m4.ma.tum.de/pers/kunz/
72. Ladneva A., Piterbarg V., On double extremes of Gaussian processes. EU RANDOM, Technische Universiteit Eindhoven, The Netherlands; document is available at http://www.eurandom.tue.nl (see eurandom reports/2000027)
73. Güttinger W., Eikemeier H., eds, Structural stability in physics. Springer, Berlin Heidelberg New York (1980)
74. Albeverio S., Lütkebohmert E., in preparation