So finally, after some cancellations, which implies (26). To establish the rest of the statements of Theorem 1 we will need the following lemma. Lemma A3. Suppose that our null hypothesis holds. Denote ∑ p j =1 λ 2 j as T. Then, for any fixed r and ¯ h < √$_{c,}$ and any ( h$_{1}$, ..., h$_{r}$ ) ∈ ( 0 , ¯ h ] r , as n, p → ∞ so that p/n → c , the vector ( S − p, T − (1 + c$_{p}$ ) p, Δ$_{p}$ ( z$_{10}$ ) , ..., Δ$_{p}$ ( z$_{r}$$_{0}$ )) converges in distribution to a Gaussian vector ( η, ζ, ξ$_{1}$, ..., ξ$_{r}$ ) with Proof: The proof of the lemma is similar to that of Lemma 12 in OMH. The convergence to the Gaussian distribution follows from Theorem 1.1 of Bai and Silverstein (2004). The formulas for the means, variances and covariances of η and ξ$_{j}$ are obtained using Theorem 1.1 iii) of Bai and Silverstein (2004) similarly to how the corresponding formulas in Lemma 12 of OMH are obtained using Theorem 1.1 ii). Therefore, below we only derive the formulae for the mean, variance, and covariances that involve ζ . Variable ζ does not appear in Lemma 12 of OMH because the lemma does not study the asymptotics of T − (1 + c$_{p}$ ) p. The fact that E ζ = 0 follows directly from Theorem 1.1 iii) of Bai and Silver- 33 E ( η ) = E ( ζ ) = E ( ξ$_{i}$ ) = 0 , Var ( η ) = c, Var ( ζ ) = 2 c ( 2 + 5 c + 2 c $^{2}$) , Cov ( η, ζ ) = 2 c (1 + c ) , Cov ( η, ξ$_{i}$ ) = − h$_{i}$, Cov ( ζ, ξ$_{i}$ ) = − h$_{i}$ ( h$_{i}$ + 2 + 2 c ) , and Cov ( ξ$_{i}$, ξ$_{k}$ ) = − ln (1 − h$_{i}$h$_{k}$/c ) L ( h ; µ ) = e 1 2 c $^{$_{p}$(}$∑ r j $_{=1}$h$_{j}$ ) 2 e − ∑ r j =1 h j c p ( S − p $^{)}$e − r ∑ i =1 Δ$_{p}$( z$_{i}$$_{0}$ ) r ∏ t =1 ( 1 − h 2 t c$_{p}$ ) 1 / 2 r ∏ i>j ( 1 − h$_{i}$h$_{j}$ c$_{p}$ ) ( 1 + O$_{p}$ ( n − $^{1}$)) ,