Extremal limit theorems for observations separated by random power law waiting times

Journal of Statistical Planning and Inference

Volumn 139, Issue 7, 2009, Pages 2175-2188

  Meerschaert, Mark M ^a   Stoev, Stilian A ^b  

^a MICHIGAN STATE UNIVERSITY   (United States)

^b UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Extremal process; Renewal process; Subordinator

Indexed keywords

EID: 62049084631     PISSN: 03783758     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jspi.2008.10.005     Document Type: Article

References (59)

1. Baeumer B., and Meerschaert M. Stochastic solutions for fractional Cauchy problems. Fractional Calculus Appl. Anal. 4 (2001) 481-500
2. Baeumer B., and Meerschaert M.M. Fractional diffusion with two time scales. Physica A 373 (2007) 237-251
3. Baeumer B., Meerschaert M.M., Benson D.A., and Wheatcraft S.W. Subordinated advection-dispersion equation for contaminant transport. Water Resour. Res. 37 (2001) 1543-1550
4. Barabási A.-L. The origin of bursts and heavy tails in human dynamics. Lett. Nature 435 (2005) 207-211
5. Becker-Kern P., Meerschaert M.M., and Scheffler H.P. Limit theorem for continuous time random walks with two time scales. J. Applied Probab. 41 (2003) 455-466
6. Bening V.E., and Korolev V.Yu. Generalized Poisson Models and their Applications in Insurance and Finance Modern Probability and Statistics (2002), VSP, Utrecht
7. Benson D., Wheatcraft S., and Meerschaert M. Application of a fractional advection-dispersion equation. Water Resour. Res. 36 (2000) 1403-1412
8. Benson D., Schumer R., Meerschaert M., and Wheatcraft S. Fractional dispersion, Lévy motions, and the MADE tracer tests. Transport Porous Media 42 (2001) 211-240
9. Benson D.A., Schumer R., and Meerschaert M.M. Recurrence of extreme events with power-law interarrival times. Geophys. Res. Lett. 34 (2007) L16404-L16408
10. Berhondo A.L.M., Taboada R.E.R., and Larralde L.A. Waiting time distribution of emissions in complex coronal mass ejections. Astrophys. Space Sci. 302 (2006) 213-216
11. Bertoin J. Lévy Processes (1996), Cambridge University Press, Cambridge
12. Billingsley P. Probability and Measure (1995), Wiley, New York
13. Bingham N.H. Limit theorems for occupation times of Markov processes. Z. Warsch. verw. Geb. 17 (1971) 1-22
14. Bondesson L., Kristiansen G., and Steutel F. Infinite divisibility of random variables and their integer parts. Statist. Probab. Lett. 28 (1996) 271-278
15. Brockmann D., Hufnagel L., and Geisel T. The scaling laws of human travel. Nature 439 (2006) 462-465
16. Caputo M. Linear models of dissipation whose Q is almost frequency independent, Part II. Geophys. J. Roy. Astron. Soc. 13 (1967) 529-539
17. Cheng B., Epstein R.I., Guyer R.A., and Young A.C. Earthquake-like behavior of soft γ-ray repeaters. Nature 382 (1995) 518-520
18. Dwass M. Extremal processes. Ann. Math. Statist. 35 4 (1964) 1718-1725
19. Embrechts P., Klüppelberg C., and Mikosch T. Modelling Extreme Events (1997), Springer, New York
20. Feller, W., 1971. An Introduction to Probability Theory and its Applications, second ed., vol. II. Wiley, New York.
21. Korolev V.Yu. Convergence of random sequences with independent random indices I. Theory Probab. Appl. 39 (1994) 282-297 (English translation)
22. Korolev V.Yu. Convergence of random sequences with independent random indices II. Theory Probab. Appl. 40 (1995) 770-772 (English translation)
23. Kruglov V.M., and Korolev V.Yu. Limit Theorems for Random Sums (1990), Moscow State University Publishing House, Moscow (in Russian)
24. Lamperti J. On extreme order statistics. Ann. Math. Statist. 35 (1964) 1726-1737
25. Lavergnat J., and Golé P. A stochastic raindrop time distribution model. J. Appl. Meteorol. 37 (1998) 805-818
26. Leadbetter M.R., Lindgren G., and Rootzén H. Extremes and Related Properties of Random Sequences and Processes (1983), Springer, New York
27. LePage R., Woodroofe M., and Zinn J. Convergence of a stable distribution via order statistics. Ann. Probab. 9 (1981) 624-632
28. Mainardi F., and Gorenflo R. On Mittag-Leffer-type functions in fractional evolution processes. J. Comput. Appl. Math. 118 (2000) 283-299
29. Mainardi F., Raberto M., Gorenflo R., and Scalas E. Fractional calculus and continuous-time finance II: the waiting-time distribution. Phys. A 287 (2000) 468-481
30. Meerschaert M., and Scheffler H.P. Limit Distributions for Sums of Independent Random Vectors (2001), Wiley, New York
31. Meerschaert M., Benson D., Scheffler H.P., and Baeumer B. Stochastic solution of space-time fractional diffusion equations. Phys. Rev. E 65 (2002) 1103-1106
32. Meerschaert M.M., and Scheffler H.P. Limit theorems for continuous time random walks with infinite mean waiting times. J. Appl. Probab. 41 3 (2004) 623-638
33. Meerschaert M.M., and Scheffler H.P. Triangular array limits for continuous time random walks. Stochastic Process. Appl. 118 9 (2008) 1606-1633
34. Metzler R., and Klafter J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A 37 (2004) R161-R208
35. Miller K., and Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley, New York
36. Pancheva, E.I., 1998. Self-similar extremal processes. in: Proceedings of the 18th Seminar on Stability Problems for Stochastic Models Part I, Hajdúszoboszló, 1997. J. Math. Sci. (New York) 92 (3), 3911-3920.
37. Pancheva E.I., and Jordanova P.K. Functional transfer theorems for maxima of iid random variables functional transfer theorems for maxima of iid random variables. Dokl. Balgaskata Akad. Nauk. C.R. Acad. Bulgare Sci. 57 8 (2004) 9-14
38. Pancheva E.I., Mitov I., and Volkovich Z. Sums and extremal processes over explosion area. Dokl. Balgaskata Akad. Nauk. C.R. Acad. Bulgare Sci. 57 8 (2006) 9-14
39. Pancheva E.I., Kolkovska E.T., and Jordanova P.K. Random time-changed extremal processes. Teor. Veroyatn. Primen. 59 12 (2006) 1219-1226 (Translated in Theory Probab. Appl.)
40. Pancheva E., Volkovich Z., and Morozensky L. Upper and lower bounds for ruin probability. Pliska Stud. Math. Bulgar. 18 (2007) 315-326
41. Podlubny I. Fractional Differential Equations (1999), Academic Press, New York
42. Resnick S.I. Inverses of extremal processes. Adv. in Appl. Probab. 6 (1974) 392-406
43. Resnick S.I. Weak convergence to extremal processes. Ann. Probab. 3 (1975) 951-960
44. Resnick S.I. Point processes, regular variation, and weak convergence. Adv. in Appl. Probab. 18 1 (1986) 66-138
45. Resnick S.I. Extreme Values, Regular Variation and Point Processes (1987), Springer, New York
46. Resnick S.I., and Rubinovitch M. The structure of extremal processes. Adv. in Appl. Probab. 5 (1973) 287-307
47. Resnick S.I., and Stǎricǎ C. Consistency of Hill's estimator for dependent data. J. Appl. Probab. 32 (1995) 139-167
48. Sabatelli L., Keating S., Dudley J., and Richmond P. Waiting time distributions in financial markets. Eur. Phys. J. B 27 (2002) 273-275
49. Samko S., Kilbas A., and Marichev O. Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach, London
50. Samorodnitsky G., and Taqqu M. Stable non-Gaussian Random Processes (1994), Chapman & Hall, New York
51. Sato K.I. Lévy Processes and Infinitely Divisible Distributions (1999), Cambridge University Press, Cambridge
52. Scalas, E., 2006. Five years of continuous-time random walks in econophysics. The Complex Networks of Economic Interactions, Lecture Notes in Economics and Mathematical Systems, vol. 567, Springer, Berlin, pp. 3-16.
53. Schumer R., Benson D.A., Meerschaert M.M., and Wheatcraft S.W. Eulerian derivation of the fractional advection-dispersion equation. J. Contaminant Hydrol. 48 (2001) 69-88
54. Seneta, E., 1976. Regularly Varying Functions. Lecture Notes in Mathematics, vol. 508, Springer, Berlin.
55. Silvestrov, D.S., 2004. Limit Theorems for Randomly Stopped Stochastic Processes. Springer Series in Statistics. Probability and its Applications, Springer, New York.
56. Silvestrov D.S., and Teugels J.L. Limit theorems for extremes with random sample size. Adv. in Appl. Probab. 30 (1998) 777-806
57. Silvestrov D.S., and Teugels J.L. Limit theorems for mixed max-sum processes with renewal stopping. Ann. Appl. Probab. 14 (2004) 1838-1868
58. Smethurst D.P., and Williams H.C. Are hospital waiting lists self-regulating?. Nature 410 (2001) 652-653
59. Tweedie M.C.K. Statistical properties of inverse Gaussian distributions I. Ann. Math. Statist. 28 (1957) 362-377