Laïp Siks Thesis

Example 5.7. (More on empirical processes) Sometimes, the X_{n} take values in a subset D ⊂ l∞ (F) admitting a Polish topology. If the X_{n} are also measurable and converge in distribution under such topology, something more can be said. To be concrete, suppose X = [0, 1] and F = {I[0,t] : 0 ≤ t ≤ 1}. Let D be the set of real cadlag functions on [0, 1], D the ball σ-field on D with respect to uniform distance, and

X_{n}(t) := X_{n}(I_{[0,t]}) = \sqrt{n}(\frac{1}{n}\sum_{i=1}^{n}I{ξi ≤t} − P (ξ1 ≤ t)), t ∈ [0, 1].

Then, X_{n} : Ω → D and X_{n}^{-1} (D) ⊂ A. If D is equipped with Skorohod topology, the Borel σ-field on D is D and X_{n} converges in distribution to a probability measure µ on D. Since D is Polish under Skorohod topology and µ{x} = 0 for all x ∈ D (unless ξ1 has a degenerate distribution, in which case everything is trivial)