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