On Linnik's continuous-time random walks

Journal of Physics A: Mathematical and General

Volumn 33, Issue 14, 2000, Pages 2631-2652

  Huillet, Thierry ^a,b  

^a CNRS   (France)

^b UNIVERSITÉ DE CERGY PONTOISE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034646495     PISSN: 03054470     EISSN: None     Source Type: Journal    
DOI: 10.1088/0305-4470/33/14/303     Document Type: Article

References (36)

1. Bertoin J 1996 Lévy Processes (Cambridge: Cambridge University Press)
2. Christoph G and Schreiber K 1999 Scaled Sibuya distribution and discrete self decomposability Magdeburg University preprint submitted
3. Christoph G and Schreiber K 1999 Positive Linnik and discrete Linnik distributions Magdeburg University preprint submitted
4. Christoph G and Schreiber K 1998 The generalized discrete Linnik distributions Advances in Stochastic Models for Reliability, Quality and Safety ed W Kahle, E v Collani, J Franz and U Jensen (Basle: Birkhäuser) pp 3-18
5. Christoph G and Schreiber K 1998 Discrete stable random variables Stat. Prob. Lett. 37 243-7
6. Cinlar E 1975 Introduction to Stochastic Processes (New York: Prentice-Hall)
7. Devroye L 1990 A note on Linnik distribution Stat. Prob. Lett. 9 305-6
8. Devroye L 1993 A triptych of discrete distributions related to stable law Stat. Prob. Lett. 18 349-51
9. Embrechts P, Klüppelberg C and Mikosh T 1997 Applications of mathematics Modelling Extremal Events vol 33 (Berlin: Springer)
10. Feller W 1971 An Introduction to Probability Theory and its Applications vol 2 (New York: Wiley)
11. Feller W 1944 On the integral equation of renewal theory Ann. Math. Stat. 12 247-67
12. Gutenberg B and Richter C F 1944 Frequency of earthquakes in California Bull. Seism. Soc. Am. 34 185-8
13. Hughes B D 1995 Random Walks and Random Environments vol 1 (Oxford: Oxford University Press)
14. Huillet T and Raynaud H-F 1999 Modelling extremal events using Gnedenko distributions J. Phys. A: Math. Gen. 32 1099-113
15. Huillet T and Raynaud H-F 1999 Rare events in a log-Weibull scenario: application to earthquake magnitude data Eur. Phys. J. B 12 457-69
16. Huillet T and Raynaud H-F 2000 On rare and extreme events Chaos Solitons Fractals at press
17. Huillet T 1999 Fractional Lévy motions and related processes J. Phys. A: Math. Gen. 32 7225-48
18. Jaffard S 1999 The multifractal nature of Lévy processes Prob. Theory Relat. Fields 114 207-27
19. Kagan Y Y 1994 Observational evidence for earthquakes as a nonlinear dynamic process Physica D 160-92
20. Kotulski M and Weron K 1996 Random walk approach to relaxation in disordered systems (Lecture Notes in Statistics vol 114) ed C C Heyde, Yu V Prohorov, R Pyke and S T Rachev (New York: Springer) pp 379-88
21. Kotulski M 1995 Asymptotic distributions of the continuous time random walks: a probabilistic approach J. Stat. Phys. 81 777-92
22. Klafter J and Schlesinger M F 1994 Probability distributions for continuous-time random walks with long tails J. Phys. Chem. 98 7366-70
23. Laherrère J and Sornette D 1998 Stretched exponential distributions in nature and economy: fat tails with characteristic scales Eur. Phys. J. B 2 525-39
24. Linnik Yu V 1963 Ukr. Math. Zh. 5 207-43, 247-90 [Engl. Transl. 1963 Linear forms and statistical criteria I, II Select. Transl. Math. Stat. Prob. 3 1-90]
25. Linnik Yu V 1963 Ukr. Math. Zh. 5 207-43, 247-90 [Engl. Transl. 1963 Linear forms and statistical criteria I, II Select. Transl. Math. Stat. Prob. 3 1-90]
26. Lukacs E 1983 Developments in Characteristic Function Theory (Griffin)
27. Mandelbrot B B 1983 The Fractal Geometry of Nature (New York: W H Freeman)
28. Montroll E W and Weiss G H 1965 Random walks on lattices II J. Math. Phys. 6 167-81
29. Pillai R N 1990 On Mittag-Leffler functions and related distributions Ann. Inst. Stat. Math. 42 157-61
30. Skrokhod A V and Gikhman I 1969 Introduction to the Theory of Random Processes (London: Saunders)
31. Sornette D and Knopoff L 1997 The paradox of the expected time until the next earthquake Bull. Seism. Soc. Am. 789-98
32. Smith W L 1955. Regenerative stochastic processes Proc. R. Soc. A 232 5-31
33. Smith W L 1958 Renewal theory and its ramifications J. R. Soc. B 20 243-84
34. Steutel F W and van Harn K 1979 Discrete analogues of self-decomposability and stability Ann. Prob. 7 893-9
35. Weissmann H, Weiss G H and Havlin S 1989 Transport properties of the continuous-time random walk with a long-tailed waiting time density J. Stat. Phys. 57 301-16
36. Weron K and Kotulski M 1996 On the Cole-Cole relaxation function and related Mittag-Leffler distribution Physical A 232 180-8