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

M. P. Etienne () and P. Vallois ()
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M. P. Etienne: Laboratoire Statistique et Genome
P. Vallois: Université Henri Poincaré

Methodology and Computing in Applied Probability, 2004, vol. 6, issue 3, 255-275

Abstract: 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 |.

Keywords: Skorokhod's embedding; random walk; local score; maximum (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1023/B:MCAP.0000026559.87023.ec

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