Weak convergence of Vervaat and Vervaat Error processes of long-range dependent sequences

We assume that the sequence ck, k ≥ 0, is regularly varying with index −β, β ∈ (1/2, 1) (written as ck ∈ RV−β). This means that ck ∼ k L0(k) as k → ∞, where L0 is slowly varying at infinity (see e.g. [2, Sections 1.4, 1.5] for the definition of slowly varying functions). We shall refer to all such models as long range dependent (LRD) linear processes. In particular, by the Karamata Theorem, the covariances ρk := EX0Xk decay at the hyperbolic School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6 Canada, email: [email protected] School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia, email: [email protected] and Mathematical Institute, Wroc law University, Pl. Grunwaldzki 2/4, 50-384 Wroc law, Poland. Research supported in part by NSERC Canada Discovery Grants of Miklós Csörgő, Donald Dawson and Barbara Szyszkowicz at Carleton University