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, Proof. Note that 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 1 d$_{n,m}$ R$_{n}$ ( x, t ) ⇒ J$_{m}$ (Ψ$_{x}$ ◦ σ ) m ! Z$_{m}$ ( t ) in D ([ −∞ , ∞ ] × [0 , 1]) . E ( $^{1}${ ψ ( X$_{j}$ ) ≤ x } | F j − 1 ) = E ( $^{1}${ ψ ( σ ( Y$_{j}$ ) ε$_{j}$ ) ≤ x } | F j − 1 ) = Ψ$_{x}$( σ ( Y$_{j}$ )) R$_{n}$ ( x, t ) = ⌊ nt ⌋ ∫ Ψ$_{x}$( u ) d ( $^{G}$⌊ nt ⌋ − E $^{G}$⌊ nt ⌋ ) ( u ) , $_{d}$− 1 $_{n,m}$R$_{n}$ ( x, t ) = − { ∫ d du P ( ψ ( uε$_{1}$ ) ≤ x ) $_{d}$− 1 n,m    ⌊ nt ⌋ ( $^{G}$⌊ nt ⌋ ( u ) − E $^{G}$⌊ nt ⌋ ( u ) ) − J$_{m}$ ( σ ; u ) m ! ⌊ nt ⌋ ∑ j =1 H$_{m}$ ( Y$_{j}$ )   $^{du}$ } − { ∫ d du P ( ψ ( uε$_{1}$ ) ≤ x ) $_{d}$− 1 n,m J$_{m}$ ( σ ; u ) m ! ⌊ nt ⌋ ∑ j =1 H$_{m}$ ( Y$_{j}$ ) du } =: I$_{1}$ ( x, t ) + I$_{2}$ ( x, t ) , where m denotes the Hermite rank of the class { $^{1}${ σ ( Y 1 ) ≤ x } − F$_{σ}$$_{(}$$_{Y}$$_{1}$$_{)}$ ( x ) , x ∈ R } and J$_{m}$ ( σ ; y ) = E ( $^{1}${ σ ( Y 1 ) ≤ y } H$_{m}$ ( Y$_{1}$ ) ) . 16