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Asymptotic normality and efficiency of the maximum likelihood estimator for the parameter of a ballistic random walk in a random environment

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Abstract

We consider a one-dimensional ballistic random walk evolving in a parametric independent and identically distributed random environment. We study the asymptotic properties of the maximum likelihood estimator of the parameter based on a single observation of the path till the time it reaches a distant site. We prove asymptotic normality for this consistent estimator as the distant site tends to infinity and establish that it achieves the Cramér-Rao bound. We also explore in a simulation setting the numerical behavior of asymptotic confidence regions for the parameter value.

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References

  1. O. Adelman and N. Enriquez, “Random Walks in Random Environment: What a Single Trajectory Tells”, Israel J. Math. 142, 205–220 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  2. I. Benjamini and H. Kesten, “Distinguishing Sceneries by Observing the Scenery along a Random Walk Path”, J.Anal.Math. 69, 97–135 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  3. H. J. Bierens, Introduction to the Mathematical and Statistical Foundations of Econometrics, in Cambridge Books (Cambridge Univ. Press, Cambridge, 2005).

    Google Scholar 

  4. P. Billingsley, Convergence of ProbabilityMeasures (Wiley, New York, 1968).

    Google Scholar 

  5. A. A. Chernov, “Replication of a Multicomponent Chain by the Lightning Mechanism”, Biofizika 12, 297–301 (1967)

    Google Scholar 

  6. F. Comets, M. Falconnet, O. Loukianov, D. Loukianova, and C. Matias, “Maximum Likelihood Estimator Consistency for Ballistic Random Walk in a Parametric Random Environment”, Stochastic Processes and Applications 124, 268–288 (2014).

    Article  MATH  MathSciNet  Google Scholar 

  7. P. Hall and C. C. Heyde, Martingale Limit Theory and Its Application, in Probab. and Math. Statist. (Academic Press, New York, 1980).

    Google Scholar 

  8. B. D. Hughes, Random Walks and Random Environments, in Oxford Science Publications. Random Environments. (The Clarendon Press, Oxford Univ. Press, New York, 1996), Vol. 2.

    Google Scholar 

  9. H. Kesten, M. V. Kozlov, and F. Spitzer, “A Limit Law for Random Walk in a Random Environment”, Compositio Math. 30, 145–168 (1975).

    MATH  MathSciNet  Google Scholar 

  10. M. V. Kozlov, “Random Walk in a One-Dimensional Random Medium”, Theory Probab. Appl. 18, 387–388, (1973).

    Article  MATH  Google Scholar 

  11. M. Löwe and Heinrich III Matzinger, “Scenery Reconstruction in Two Dimensions with Many Colors”, Ann. Appl. Probab. 12, 1322–1347 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  12. H. Matzinger, “Reconstructing a Three-Color Scenery by Observing It along a Simple Random Walk Path”, Random Structures Algorithms 15, 196–207 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  13. D. Revuz, Markov chains, in North-Holland Mathematical Library (North-Holland Publ. Co., Amsterdam, 1984), Vol. 11, 2nd ed.

    Google Scholar 

  14. F. Solomon, “Random Walks in a Random Environment”, Ann. Probab. 3, 1–31 (1975).

    Article  MATH  Google Scholar 

  15. D. E. Temkin, “One-Dimensional Random Walks in a Two-Component Chain”, Soviet Math. Doklady 13, 1172–1176 (1972).

    MATH  Google Scholar 

  16. O. Zeitouni, Random Walks in Random Environment, in Lecture Notes in Math., Vol. 1837: Lectures on Probab. Theory and Statist. (Springer, Berlin, 2004), pp. 189–312.

    Google Scholar 

Download references

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Correspondence to M. Falconnet.

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Falconnet, M., Loukianova, D. & Matias, C. Asymptotic normality and efficiency of the maximum likelihood estimator for the parameter of a ballistic random walk in a random environment. Math. Meth. Stat. 23, 1–19 (2014). https://doi.org/10.3103/S1066530714010013

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  • DOI: https://doi.org/10.3103/S1066530714010013

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