content_image/306261.jpg content_image/306260.jpg content_image/306262.jpg
Consider now the original matrix G ( y ) , denote it G $^{1}$( y ) with q$_{1}$ = q and employ the construction recursively until for some m it ends: Σ m v $_{=1}$¯ r$_{v}$ = q. 

Denote by ¯ S v the matrix    I$_{¯}$ r$_{1}$ + ... +¯ r$_{v}$$_{−}$$_{1}$ S v    and define S = ¯ S $^{m}$... ¯ S $^{1}$. Set a = ( α$_{1}$, ..., α$_{q}$ ) = ( ¯ k$_{1}$, ..., ¯ k$_{1}$, ..., ¯ k$_{m}$, ..., ¯ k$_{m}$ ) , where each ¯ k$_{v}$ enters ¯ r$_{v}$ times. Then for this a and S 

is a finite matrix ¯ G ( y ) =    ¯ G $^{1}$( y ) . ¯ G $^{m}$( y )   $^{; if}$ q ∑ i =1 α$_{i}$ = ¯ α, then CLDR property holds, if q α$_{i}$ < ¯ α then the limit matrix has deficient rank. 

∑ i =1 Proof of Theorem 4.1. Consider y ∗ T = λ$_{T}$ y$_{T}$ 

and the quadratic form similar to (21) 

<img src='content_image/306261.jpg'>

<img src='content_image/306260.jpg'>

From Assumption 2.3 if λ = λ$_{T}$ and θ = ˆ θ$_{T}$ then the probability limit of corresponding V − 1 2 A − $^{1}$y ∗ T is Z with distribution Q ( ¯ θ ) ; from Assumption 2.4 ˆ V$_{T}$ = V + o$_{p}$ (1) . From (20) and convergence it follows that 

<img src='content_image/306262.jpg'>

