and The integer m is called the Hermite rank of g and we refer to J$_{q}$ ( g ) as the q -th Hermite coefficient of g . We will also consider the Hermite expansion of the function class { $^{1}${ g ( Y 1 ) ≤ x } − F$_{g}$$_{(}$$_{Y}$$_{1}$$_{)}$ ( x ) , x ∈ R } , where F$_{g}$$_{(}$$_{Y}$$_{1}$$_{)}$ denotes the distribution function of g ( Y$_{1}$ ). For fixed x , we have with J$_{q}$ ( g ; x ) = E ( $^{1}${ g ( Y 1 ) ≤ x } H$_{q}$ ( Y$_{1}$ ) ) . The Hermite rank corresponding to this function class is defined by m = inf$_{x}$ m ( x ), where m ( x $^{) denotes the Hermite rank of 1}${ g ( Y 1 ) ≤ x } − F$_{g}$$_{(}$$_{Y}$$_{1}$$_{)}$ ( x ). We refer to Beran et al. (2013) for further details. The asymptotic behavior of partial sums of subordinated Gaussian sequences is char- acterized in Taqqu (1979). Due to the functional non-central limit theorem in that paper, where Z$_{m}$ ( t ), 0 ≤ t ≤ 1, is an m -th order Hermite process, and the convergence holds in D ([0 , 1]), the space of functions that are right-continuous with left limits. In fact, the limiting behavior in (5) is the same as that of the corresponding partial sums based on { H$_{m}$ ( Y$_{j}$ ) , j ≥ 1 } : Moreover, the functional central limit theorem for the empirical processes was established $^{1}${ g ( Y 1 ) ≤ x } − F$_{g}$$_{(}$$_{Y}$$_{1}$$_{)}$ ( x ) = ∞ ∑ q = m J$_{q}$ ( g ; x ) q ! H$_{q}$ ( Y$_{1}$ ) m = inf { q ≥ 1 | J$_{q}$ ( g ) = 0 } . 1 d$_{n,m}$ ⌊ nt ⌋ ∑ j =1 g ( Y$_{j}$ ) ⇒ J$_{m}$ ( g ) m ! Z$_{m}$ ( t ) , 0 ≤ t ≤ 1 , (5) d 2 n,m = Var ( n ∑ j =1 H$_{m}$ ( Y$_{j}$ ) ) ∼ c$_{m}$n 2 − $^{mD}$L$^{m}$( n ) , c$_{m}$ = 2 m ! (1 − Dm )(2 − Dm ) , J$_{m}$ ( g ) 1 d$_{n,m}$ ⌊ nt ⌋ ∑ j =1 H$_{m}$ ( Y$_{j}$ ) ⇒ J$_{m}$ ( g ) m ! Z$_{m}$ ( t ) , 0 ≤ t ≤ 1 . (6) 8