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Approximation of the Distribution of the Supremum of a Centered Random Walk. Application to the Local Score

Author

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  • M. P. Etienne

    (Laboratoire Statistique et Genome)

  • P. Vallois

    (Université Henri Poincaré)

Abstract

Let (X n ) n ≥ 0 be a real random walk starting at 0, with centered increments bounded by a constant K. The main result of this study is: |P(S n √ n ≥ x)−P(σ sup0 ≤ u ≤ 1 B u ≥ x)|≤ C(n,K)√ ∈ n/n, where x ≥ 0, σ2 is the variance of the increments, S n is the supremum at time n of the random walk, (B u ,u≥ 0) is a standard linear Brownian motion and C(n,K) is an explicit constant. We also prove that in the previous inequality S n can be replaced by the local score and sup0 ≤ u ≤ 1 B u by sup0 ≤ u ≤ 1|B u |.

Suggested Citation

  • M. P. Etienne & P. Vallois, 2004. "Approximation of the Distribution of the Supremum of a Centered Random Walk. Application to the Local Score," Methodology and Computing in Applied Probability, Springer, vol. 6(3), pages 255-275, September.
    • Handle: RePEc:spr:metcap:v:6:y:2004:i:3:d:10.1023_b:mcap.0000026559.87023.ec
    DOI: 10.1023/B:MCAP.0000026559.87023.ec
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    References listed on IDEAS

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    1. Daudin, Jean-Jacques & Etienne, Marie Pierre & Vallois, Pierre, 2003. "Asymptotic behavior of the local score of independent and identically distributed random sequences," Stochastic Processes and their Applications, Elsevier, vol. 107(1), pages 1-28, September.
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    Cited by:

    1. Chabriac, Claudie & Lagnoux, Agnès & Mercier, Sabine & Vallois, Pierre, 2014. "Elements related to the largest complete excursion of a reflected BM stopped at a fixed time. Application to local score," Stochastic Processes and their Applications, Elsevier, vol. 124(12), pages 4202-4223.

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