Functional central limit theorems for sums of nearly nonstationary processes*

We study some Holderian functional central limit theorems for the polygonal partial-sum processes built on a first-order autoregressive process yn,k = φnyn,k−1 + εk with ϕn converging to 1 and i.i.d. centered square-integrable innovations. In the case where ϕn = eγ/n with a negative constant γ, we prove that the limiting process is an integrated Ornstein–Uhlenbeck one. In the case where ϕn = 1− γn/n, with γn tending to infinity slower than n, the convergence to Brownian motion is established in Holder space in terms of the rate of γn and the integrability of the εks.