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

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|.