content_image/299573.jpg content_image/299574.jpg content_image/299575.jpg content_image/299576.jpg content_image/299578.jpg
material. 

Long memory part. Finally, we prove weak convergence of the long memory part R$_{n}$ ( x, t ). 

Lemma 3.7. Under the conditions of Theorem 3.3, 

<img src='content_image/299573.jpg'>

Proof. Note that 

<img src='content_image/299574.jpg'>

because Y$_{j}$ is F j − $_{1}$-measurable and ε$_{j}$ is independent of F j − $_{1}$. Furthermore, E (Ψ$_{x}$ ( σ ( Y$_{j}$ ))) = F$_{ψ}$$_{(}$$_{X}$$_{1}$$_{)}$ ( x ), where F$_{ψ}$$_{(}$$_{X}$$_{1}$$_{)}$ denotes the distribution function of ψ ( X$_{1}$ ) = ψ ( σ ( Y$_{1}$ ) ε$_{1}$ ). Hence, 

where G$_{l}$ ( u ) = 1 l ∑ l j $^{$_{=1}$1}${ σ ( Y$_{j}$ ) ≤ u } is the empirical distribution function of the sequence { σ ( Y$_{j}$ ) , j ≥ 1 } . We have 

<img src='content_image/299575.jpg'>

<img src='content_image/299576.jpg'>

where m denotes the Hermite rank of the class { $^{1}${ σ ( Y 1 ) ≤ x } − F$_{σ}$$_{(}$$_{Y}$$_{1}$$_{)}$ ( x ) , x ∈ R } and 

<img src='content_image/299578.jpg'>

