Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T22:22:03.880Z Has data issue: false hasContentIssue false
  • English
  • Français

The extremes of random walks in random sceneries

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Brice Franke  and
Tatsuhiko Saigo
Brice Franke*
Affiliation:
Ruhr-Universität Bochum
Tatsuhiko Saigo*
Affiliation:
National Taiwan University
*
∗ Postal address: Ruhr-Universität Bochum, Fakultät für Mathematik, Universitätsstr. 150, 44780 Bochum, Germany. Email address: brice.franke@rub.de
∗∗ Current address: Department of Mathematics, Keio University 3-14-1 Hiyoshi, Kohoku-ku, Yokohama-shi, Kanagawa-ken prefecture, 223-8522, Japan. Email address: saigo@math.keio.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this article we analyse the behaviour of the extremes of a random walk in a random scenery. The random walk is assumed to be in the domain of attraction of a stable law, and the scenery is assumed to be in the domain of attraction of an extreme value distribution. The resulting random sequence is stationary and strongly dependent if the underlying random walk is recurrent. We prove a limit theorem for the extremes of the resulting stationary process. However, if the underlying random walk is recurrent, the limit distribution is not in the class of classical extreme value distributions.

Keywords

Extremesstationary sequencesrandom walkrandom scenery

MSC classification

Primary: 60G70: Extreme value theory; extremal processes 60G50: Sums of independent random variables; random walks
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 2 , June 2009 , pp. 452 - 468
Copyright
Copyright © Applied Probability Trust 2009 

References

Arai, T. (2001). A class of semi-selfsimilar processes related to random walks in random scenery. Tokyo J. Math. 24, 69–85.CrossRefGoogle Scholar
Billingsley, P. (1986). Probability and Measure, 2nd edn. John Wiley, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Denzel, G. E. and O'Brien, G. (1975). Limit theorems for extreme values of chain-dependent processes. Ann. Prob. 3, 773–779.CrossRefGoogle Scholar
Fereira, H. (1998). Doubly stochastic compound Poisson processes in extreme value theory. Portugal. Math. 55, 465–474.Google Scholar
Gnedenko, B. (1943). Sur la distribution limite du terme d'une série aléatoire. Ann. Math. 44, 423–453.CrossRefGoogle Scholar
Hsing, T., Hüsler, J. and Leadbetter, M. R. (1988). On the exceedance point process for a stationary sequence. Prob. Theory Relat. Fields 78, 97–112.CrossRefGoogle Scholar
Katz, R. W. (1977). An application of chain-dependent processes to meteorology. J. Appl. Prob. 14, 598–603.CrossRefGoogle Scholar
Kesten, H. and Spitzer, F. (1979). A limit theorem related to a new class of self-similar processes. Z. Wahrscheinlichkeitsth. 50, 5–25.CrossRefGoogle Scholar
Lang, R. and Nguyen, X.-X. (1983). Strongly correlated random fields as observed by a random walker. Z. Wahrscheinlichkeitsth. 64, 327–340.CrossRefGoogle Scholar
Leadbetter, M. R. (1983). Extremes and local dependence in stationary sequences. Z. Wahrscheinlichkeitsth., 65, 291–306.CrossRefGoogle Scholar
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.Google Scholar
Le Gall, J.-F. and Rosen, J. (1991). The range of stable random walks. Ann. Prob. 19, 650–705.Google Scholar
Loynes, R. M. (1965). Extreme values in uniformly mixing stationary stochastic processes. Ann. Math. Statist. 36, 993–999.CrossRefGoogle Scholar
Maejima, M. (1996). Limit theorems related to a class of operator-self-similar processes. Nagoya Math. J. 142, 161–181.CrossRefGoogle Scholar
Newell, G. F. (1964). Asymptotic extremes for m-dependent random variables. Ann. Math. Statist. 35, 1322–1325.CrossRefGoogle Scholar
Resnick, S. I. (1972). Stability of maxima of random variables defined on a Markov chain. Adv. Appl. Prob. 4, 285–295.CrossRefGoogle Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes (Appl. Prob. 4). Springer, New York.Google Scholar
Saigo, T. and Takahashi, H. (2005). Limit theorems related to a class of operator semi-selfsimilar processes. J. Math. Sci. Univ. Tokyo 12, 111–140.Google Scholar
Shieh, N. R. (1995). Some self-similar processes related to local times. Statist. Prob. Lett. 24, 213–218.CrossRefGoogle Scholar
Spitzer, F. (1976). Principles of Random Walks, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Turkman, K. F. and Oliveira, M. F. (1992). Limit laws for the maxima of chain-dependent sequences with positive extremal index. J. Appl. Prob. 29, 222–227.CrossRefGoogle Scholar
Turkman, K. F. and Walker, A. M. (1983). Limit laws for the maxima of a class of quasistationary sequences. J. Appl. Prob. 20, 814–821.CrossRefGoogle Scholar