The rest of the paper is organized as follows. Section 2 establishes the weak convergence of the log-likelihood ratio process to a Gaussian process. Section 3 provides an analysis of the asymptotic powers of various sphericity tests, derives the asymptotic power envelope, and proves its validity for general invariant tests. Section 4 concludes. All proofs are given in the Appendix. Let X be a p × n$_{p}$ matrix with independent Gaussian N (0 , σ $^{2}$( I$_{p}$ + V HV ′ )) columns. Let λ$_{p}$$_{1}$ ≥ ... ≥ λ$_{pp}$ be the ordered eigenvalues of 1 $_{n$_{p}$}$XX ′ and write λ$_{p}$ = ( λ$_{p}$$_{1}$, ..., λ$_{pm}$ ) , where m = min { p,n$_{p}$ } . Similarly, let µ$_{pi}$ = λ$_{pi}$/ ( λ$_{p}$$_{1}$ + ... + λ$_{pp}$ ) , i = 1 , ..., m and µ$_{p}$ = ( µ$_{p}$$_{1}$$^{, ..., µ$_{p,m}$}$− 1 ) . As explained in the introduction, our goal is to study the asymptotic power, as n$_{p}$, p → c ∞ , of the eigenvalue-based tests of H$_{0}$ : h$_{1}$ = ... = h$_{r}$ = 0 against H$_{1}$ : h$_{j}$ > 0 for some i = 1 , ..., r , where h$_{j}$ are the diagonal elements of the diagonal matrix H . If σ 2 is specified, the model is invariant with respect to left and right orthogonal transformations; sufficiency and invariance arguments (see Appendix 5.4 for details) lead to considering tests based on λ$_{p}$ only. If σ 2 is unspecified, the model is invariant with respect to left and right orthogonal transformations and multiplications by non-zero scalars; sufficiency and invariance arguments (see Appendix 5.4) lead to considering tests based on µ$_{p}$ only. Note that the distribution of µ$_{p}$ does not depend on σ $^{2}$, whereas, if σ 2 is specified, we can always normalize λ$_{p}$ dividing it by σ $^{2}$. Therefore, we henceforth assume without loss of generality that σ 2 = 1. Let us denote the joint density of λ$_{p}$$_{1}$, ..., λ$_{pm}$ at ˜ x = ( x$_{1}$, ..., x$_{m}$ ) ∈ ( R $^{+}$) m as f$_{λp}$ (˜ x ; h ), and that of µ$_{p}$$_{1}$$^{, ..., µ$_{p,m}$}$− 1 at ˜ y = ( y$_{1}$$^{, ..., y$_{m}$}$− $_{1}$) ∈ ( R $^{+}$) m − 1 as f$_{µp}$ (˜ y ; h ). 6 2 Asymptotics of likelihood ratio processes