Detecting the complexity of a functional time series

Abstract

Consider the U -distribution function defined as P (d (X, Y ) ≤ h) = E I {d(X,Y )≤h} , where Y is an independent copy of X and whose empirical counterpart is φ n (h). Note that, thanks to the law of total expectation, Assumption (A-2), the identification constraint E [ψ (X)] = 1, when h is small enough it holds:Thus, to prove the asymptotic behavior of φ n (h) towards φ θ0 (h) is equivalent to show that φ n (h) converges to the U -distribution function E I {d(X,Y )≤h} . Moreover, it is useful to note that, since the monomial behaviour is assumed, φ θ0 (h) is Lipschitz continuous on H and the U -distribution as well. In fact, for h, h ∈ H, there exists a positive constant L that