Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T00:40:51.890Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for continuous-time random walks with infinite mean waiting times

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Mark M. Meerschaert  and
Hans-Peter Scheffler
Mark M. Meerschaert*
Affiliation:
University of Nevada, Reno
Hans-Peter Scheffler*
Affiliation:
University of Dortmund
*
∗ Postal address: Department of Mathematics, University of Nevada, Reno, NV 89557, USA. Email address: [email protected]
∗∗ Postal address: Fachbereich Mathematik, University of Dortmund, 44221 Dortmund, Germany. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A continuous-time random walk is a simple random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper we show that, when the time between renewals has infinite mean, the scaling limit is an operator Lévy motion subordinated to the hitting time process of a classical stable subordinator. Density functions for the limit process solve a fractional Cauchy problem, the generalization of a fractional partial differential equation for Hamiltonian chaos. We also establish a functional limit theorem for random walks with jumps in the strict generalized domain of attraction of a full operator stable law, which is of some independent interest.

Keywords

Operator self-similar processcontinuous-time random walk

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60K40: Other physical applications of random processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 623 - 638
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baeumer, B., and Meerschaert, M. M. (2001). Stochastic solutions for fractional Cauchy problems. Fractional Calculus Appl. Anal. 4, 481–500.Google Scholar
Barkai, E., Metzler, R., and Klafter, J. (2000). From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E 61, 132–138.10.1103/PhysRevE.61.132Google Scholar
Benson, D. A., Wheatcraft, S. W., and Meerschaert, M. M. (2000). Application of a fractional advection-dispersion equation. Water Resources Res. 36, 1403–1412.10.1029/2000WR900031Google Scholar
Benson, D. A., Wheatcraft, S. W., and Meerschaert, M. M. (2000). The fractional-order governing equation of Lévy motion. Water Resources Res. 36, 1413–1424.10.1029/2000WR900032Google Scholar
Benson, D. A., Schumer, R., Meerschaert, M. M., and Wheatcraft, S. W. (2001). Fractional dispersion, Lévy motion, and the MADE tracer tests. Transport Porous Media 42, 211–240.10.1023/A:1006733002131Google Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Bingham, N. H. (1971). Limit theorems for occupation times of Markov processes. Z. Warscheinlichkeitsth. 17, 1–22.10.1007/BF00538470Google Scholar
Blumen, A., Zumofen, G., and Klafter, J. (1989). Transport aspects in anomalous diffusion: Lévy walks. Phys Rev. A 40, 3964–3973.10.1103/PhysRevA.40.3964Google Scholar
Bondesson, L., Kristiansen, G., and Steutel, F. (1996). Infinite divisibility of random variables and their integer parts. Statist. Prob. Lett. 28, 271–278.10.1016/0167-7152(95)00135-2Google Scholar
Chaves, A. S. (1998). A fractional diffusion equation to describe Lévy flights. Phys. Lett. A 239, 13–16.10.1016/S0375-9601(97)00947-XGoogle Scholar
Embrechts, P., and Maejima, M. (2002). Selfsimilar Processes. Princeton University Press.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Getoor, R. K. (1961). First passage times for symmetric stable processes in space. Trans. Am. Math. Soc. 101, 75–90.10.1090/S0002-9947-1961-0137148-5Google Scholar
Gihman, I. I., and Skorohod, A. V. (1974). The Theory of Stochastic Processes. I. Springer, New York.Google Scholar
Hille, E., and Phillips, R. S. (1957). Functional Analysis and Semi-Groups (Amer. Math. Soc. Coll. Pub. 31). American Mathematical Society, Providence, RI.Google Scholar
Jurek, Z. J., and Mason, J. D. (1993). Operator-Limit Distributions in Probability Theory. John Wiley, New York.Google Scholar
Klafter, J., Blumen, A., and Shlesinger, M. F. (1987). Stochastic pathways to anomalous diffusion. Phys. Rev. A 35, 3081–3085.10.1103/PhysRevA.35.3081Google Scholar
Kotulski, M. (1995). Asymptotic distributions of the continuous time random walks: a probabilistic approach. J. Statist. Phys. 81, 777–792.10.1007/BF02179257Google Scholar
Kotulski, M., and Weron, K. (1996). Random walk approach to relaxation in disordered systems. In Athens Conference on Applied Probability and Time Series Analysis (Lect. Notes Statist. 114), Vol. 1, Springer, New York, pp. 379–388.10.1007/978-1-4612-0749-8_27Google Scholar
Kozubowski, T. J. (1999). Geometric stable laws: estimation and applications. Math. Comput. Modelling 29, 241–253.10.1016/S0895-7177(99)00107-7Google Scholar
Kozubowski, T. J., and Panorska, A. K. (1996). On moments and tail behavior of ν-stable random variables. Statist. Prob. Lett. 29, 307–315.10.1016/0167-7152(95)00187-5Google Scholar
Kozubowski, T. J., and Panorska, A. K. (1998). Weak limits for multivariate random sums. J. Multivariate Analysis 67, 398–413.10.1006/jmva.1998.1768Google Scholar
Kozubowski, T. J., and Panorska, A. K. (1999). Multivariate geometric stable distributions in financial applications. Math. Comput. Modelling 29, 83–92.10.1016/S0895-7177(99)00094-1Google Scholar
Kozubowski, T. J., and Rachev, S. T. (1999). Multivariate geometric stable laws. J. Comput. Analysis Appl. 1, 349–385.Google Scholar
Meerschaert, M. M., and Scheffler, H. P. (2001). Limit Distributions for Sums of Independent Random Vectors. John Wiley, New York.Google Scholar
Meerschaert, M. M., Benson, D., and Baeumer, B. (1999). Multidimensional advection and fractional dispersion. Phys. Rev. E 59, 5026–5028.10.1103/PhysRevE.59.5026Google Scholar
Meerschaert, M. M., Benson, D., and Baeumer, B. (2001). Operator Lévy motion and multiscaling anomalous diffusion. Phys. Rev. E 63, 1112–1117.10.1103/PhysRevE.63.021112Google Scholar
Metzler, R., and Klafter, J. (2000). The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77.10.1016/S0370-1573(00)00070-3Google Scholar
Montroll, E. W., and Weiss, G. H. (1965). Random walks on lattices. II. J. Math. Phys. 6, 167–181.10.1063/1.1704269Google Scholar
Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations (Appl. Math. Sci. 44). Springer, New York.Google Scholar
Port, S. C. (1967). Hitting times for transient stable processes. Pacific J. Math. 21, 161–165.10.2140/pjm.1967.21.161Google Scholar
Saichev, A. I., and Zaslavsky, G. M. (1997). Fractional kinetic equations: solutions and applications. Chaos 7, 753–764.10.1063/1.166272Google Scholar
Samorodnitsky, G., and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.Google Scholar
Samko, S., Kilbas, A., and Marichev, O. (1993). Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, London.Google Scholar
Sato, K. I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Scher, H., and Lax, M. (1973). Stochastic transport in a disordered solid. I. Theory. Phys. Rev. B 7, 4491–4502.10.1103/PhysRevB.7.4491Google Scholar
Seneta, E. (1976). Regularly Varying Functions (Lecture Notes Math. 508). Springer, Berlin.Google Scholar
Shlesinger, M., Klafter, J., and Wong, Y. M. (1982). Random walks with infinite spatial and temporal moments. J. Statist. Phys. 27, 499–512.10.1007/BF01011089Google Scholar
Shlesinger, M., Zaslavsky, G., and Frisch, U. (eds) (1995). Lévy Flights and Related Topics in Physics (Proc. Internat. Workshop, Nice, June 1994; Lecture Notes Phys. 450). Springer, Berlin.10.1007/3-540-59222-9Google Scholar
Stone, C. (1963). Weak convergence of stochastic processes defined on semi-infinite time intervals. Proc. Am. Math. Soc. 14, 694–696.10.1090/S0002-9939-1963-0153046-2Google Scholar
Uchaikin, V. V., and Zolotarev, V. M. (1999). Chance and Stability. Stable Distributions and Their Applications. VSP, Utrecht.10.1515/9783110935974Google Scholar
Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.10.1007/b97479Google Scholar
Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Operat. Res. 5, 67–85.10.1287/moor.5.1.67Google Scholar
Zaslavsky, G. M. (1994). Fractional kinetic equation for Hamiltonian chaos. Chaotic advection, tracer dynamics and turbulent dispersion. Physica D 76, 110–122.Google Scholar