content_image/296744.jpg content_image/296743.jpg content_image/296745.jpg content_image/296746.jpg content_image/296747.jpg
and 

<img src='content_image/296744.jpg'>

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 

<img src='content_image/296743.jpg'>

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, 

<img src='content_image/296745.jpg'>

<img src='content_image/296746.jpg'>

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 } : 

<img src='content_image/296747.jpg'>

Moreover, the functional central limit theorem for the empirical processes was established 

