has expectation zero. In the covariance the terms where l = j are zero and and thus converges to cov ( ψ$_{1}$, ψ$_{2}$ ) . Next (similarly to Zinde-Walsh, 2008) we show that for any set of linearly independent functions ψ$_{1}$, ..., ψ$_{m}$ ∈ D with E ( ψ 2 $_{l}$) > 0 the joint distribution of the vector converges to a multivariate Gaussian. Define similarly the vector − → η nhj with components ( η$_{nhj}$, ψ$_{l}$ ) . Denote by S the m × m matrix with ts component { S } ts = ( C, ( ψ$_{t}$, ψ$_{s}$ )) where the functional C is given by (9) . Denote by ˆ S$_{n}$ the covariance matrix of − → η $_{nhj}$. By the convergence results for T$_{lj}$, ˆ S$_{n}$ → Σ . Since the functions ψ$_{1}$, ..., ψ$_{m}$ are linearly independent and E ( ψ 2 $_{l}$) > 0 the matrix S and thus ˆ S$_{n}$ for large enough n is invertible. Define ξ$_{nhj}$ to equal ˆ S − 1 / 2 n − → η $_{nhj}$, then ˆ S − 1 / 2 n − → η nhj − S − 1 / 2 − → η nhj → p 0 . Next, consider an m × 1 vector λ with λ $^{′}$λ = 1 . The random variables λ $^{′}$ξ$_{nhj}$ are independent with expectation 0, var ∑ λ $^{′}$ξ$_{nhj}$ = 1; they satisfy the Liapunov condition: ∑ E ∣ λ $^{′}$ξ$_{nhj}$ ∣ 2+ δ → 0 for δ > 0 since the kernel function is bounded with finite support. Thus ∑ λ $^{′}$ξ$_{nhj}$ → d N (0 , 1) − → η nh = (( η$_{nh}$, ψ$_{1}$ ) ..., ( η$_{nh}$, ψ$_{m}$ )) ′ 36 n − $^{1}$E ( η$_{nhj}$, ψ$_{1}$ )( η$_{nhj}$, ψ$_{2}$ ) = T$_{jj}$ + O ( ¯ h ) ,