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

Authors
  • Etienne, M. P.1
  • Vallois, P.2
  • 1 Laboratoire Statistique et Genome, Tour Envy 2, 523 place des terrasses, Evry, 91034, France , Evry
  • 2 Université Henri Poincaré, Institut de Mathématiques Elie Cartan, 54506, Vandoeuvre Lès Nancy Cedex, France , Vandoeuvre Lès Nancy Cedex
Type
Published Article
Journal
Methodology And Computing In Applied Probability
Publisher
Kluwer Academic Publishers
Publication Date
Sep 01, 2004
Volume
6
Issue
3
Pages
255–275
Identifiers
DOI: 10.1023/B:MCAP.0000026559.87023.ec
Source
Springer Nature
Keywords
License
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Abstract

Let (Xn)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(Sn √ n ≥ x)−P(σ sup0 ≤ u ≤ 1Bu ≥ x)|≤ C(n,K)√ ∈ n/n, where x ≥ 0, σ2 is the variance of the increments, Sn is the supremum at time n of the random walk, (Bu,u≥ 0) is a standard linear Brownian motion and C(n,K) is an explicit constant. We also prove that in the previous inequality Sn can be replaced by the local score and sup0 ≤ u ≤ 1 Bu by sup0 ≤ u ≤ 1|Bu|.

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