Necessary and Sufficient Condition for the Functional Central Limit Theorem in Hölder Spaces

Hölderian invariance principle for Hilbertian linear processes

“…Condition (1.9) is optimal because the class of linear processes considered includes the special case where X k = k and it is known from [12] that in this case (1.9) is necessary for the weak-H o ρ (H) convergence of n −1/2 ξ n to W Q . It is easily seen that if α < 1/2 in (1.1), then we can drop the requirement "for every c > 0" in (1.9) and simply take c = 1.…”

“…It is easily seen that if α < 1/2 in (1.1), then we can drop the requirement "for every c > 0" in (1.9) and simply take c = 1. But this requirement cannot be dropped if α = 1/2, see Remark 12 in [12].…”

“…From this point, the remainder part of a detailed proof would be an exact reproduction of the corresponding part in the proof of Theorem 8 in [12], pp. 235-237.…”

“…, where j 0 is defined as in [12], p.237. It is easily seen that there is always a choice of β making compatible both conditions imposed on γ.…”

“…To discard triviality, we may assume that ρ(h) ≥ ch for some positive constant c. Then H o ρ (B) inherits the separability of B (see [11]). As in [12], we shall restrict our study to the case of weight functions ρ in the class R defined below. For any ρ in R, the space H o ρ (B) supports any B-valued Brownian motion.…”