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Stochastic stability of some state-dependent growth-collapse processes

Part of: Limit theorems Markov processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Christian Y. Robert
Christian Y. Robert*
Affiliation:
CNAM and CREST
*
∗ Postal address: Centre de Recherche en Economie et Statistique, Timbre J320, 15 Boulevard Gabriel Peri, 92245 Malakoff Cedex, France. Email address: chrobert@ensae.fr
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Abstract

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In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.

Keywords

Growth-collapse processMarkov chainlong-range dependenceheavy-tailed distribution

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 60G10: Stationary processes 60G51: Processes with independent increments; L'evy processes 60F10: Large deviations

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 1 , March 2007 , pp. 189 - 220
Copyright
Copyright © Applied Probability Trust 2007 

References

Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Bak, P. (1996). How Nature Works: The Science of Self-Organized Criticality. Copernicus, New York.CrossRefGoogle Scholar
Bak, P., Christensen, K., Danon, L. and Scanlon, T. (2002). Unified scaling law for earthquakes. Phys. Rev. Lett. 88, 178501.Google Scholar
Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
Boffetta, G. et al. (1999). Power laws in solar flares: self-organized criticality or turbulence? Phys. Rev. Lett. 83, 4662.CrossRefGoogle Scholar
Boxma, O., Perry, D., Stadje, W. and Zacks, S. (2006). A Markovian growth-collapse model. Adv. Appl. Prob. 38, 221–243.Google Scholar
Corral, A. (2004). Long-term clustering, scaling, and universality in the temporal occurrence of earthquakes. Phys. Rev. Lett. 92, 108501.Google Scholar
Dumas, V., Guillemin, F. and Robert, P. (2002). A Markovian analysis of additive-increase multiplicative-decrease algorithms. Adv. Appl. Prob. 34, 85–111.Google Scholar
Eliazar, I. and Klafter, J. (2003). A growth-collapse model: Lévy inflow, geometric crashes, and generalized Ornstein–Uhlenbeck dynamics. Physica A 334, 1–21.Google Scholar
Eliazar, I. and Klafter, J. (2005). Stochastic Ornstein–Uhlenbeck capacitors. J. Statist. Phys. 118, 177–198.Google Scholar
Lippiello, E., de Arcangelis, L. and Godano, C. (2005). Memory in self-organized criticality. Europhys. Lett. 72, 678–684.Google Scholar
Mega, M. et al. (2003). Power-law time distribution of large earthquakes. Phys. Rev. Lett. 90, 188501.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, New York.Google Scholar
Sanchez, R., Newman, D. E. and Carreras, B. A. (2002). Waiting-time statistics of self-organized criticality systems. Phys. Rev. Lett. 88, 068302.Google Scholar
Sole, R. V. and Manrubia, S. C. (1996). Extinction and self-organized criticality in a model of large-scale evolution. Phys. Rev. Lett. 54, 108501.Google Scholar
Sole, R. V. et al. (1998). Long-range correlations in the fossil records and the fractal nature of macroevolution. Adv. Complex Systems 1, 255–266.CrossRefGoogle Scholar
Sznajd-Weron, K. and Weron, R. (2001). A new model of mass extinctions. Physica A 293, 559–565.Google Scholar
Taqqu, M. (2003). Fractional Brownian motion and long-range dependence. In Theory of Long-Range Dependence, eds Doukhan, P., Oppenheim, G. and Taqqu, M. S., Birkhäuser, Boston, MA, pp. 5–38.Google Scholar
Wheatland, M. S. (2000). The origin of the solar flare waiting-time distribution. Astrophys. J. 536, L109–L112.Google Scholar
Wheatland, M. S. and Litvinenko, Y. E. (2002). Understanding solar flare waiting-time distributions. Solar Phys. 211, 255–274.Google Scholar
Woodard, R., Newman, D. E., Sanchez, R. and Carreras, B. A. (2004). Comment on “Do earthquakes exhibit self-organized criticality?”. Phys. Rev. Lett. 93, 249801.Google Scholar