tation to obtain the asymptotics of the likelihood ratio process. This asymptotics is then used along with the Neyman-Pearson lemma and Le Cam’s third lemma to establish a simple analytical formula for the maximal possible asymptotic probabil- ity of correct signal detection holding the asymptotic probability of false detection fixed. Section 6 concludes. All proofs are relegated to the Appendix. As mentioned above, we assume that data consist of n independent observations of p -dimensional complex-valued Gaussian vectors X$_{t}$ ∼ N C (0 , Σ). This means that X$_{t}$ = Y$_{t}$ + i Z$_{t}$ , where i denotes the imaginary unit, and the joint density of ( Y$_{t}$, Z$_{t}$ ) at ( y, z ) equals 1 (2 π ) $^{p}$det Σ exp { − tr [ Σ − $^{1}$( y + i z ) ( y − i z ) $^{′}$]} (see, for exam- ple, Goodman, 1963). Further, we assume that the covariance matrix Σ equals σ $^{2}$( I$_{p}$ + V H ¯ V $^{′}$) , where H = diag ( h$_{1}$, ..., h$_{r}$ ) quantifies the sizes of the covariance spikes. Our goal is to study the asymptotic power of tests of H$_{0}$ : h$_{1}$ = ... = h$_{r}$ = 0 against H$_{1}$ : h$_{i}$ > 0 for some i = 1 , ..., r . If σ 2 is specified, the model is invariant with respect to unitary transformations and the maximal invariant statistic is λ , the vector of the first m = min { n, p } eigen- values of XX $^{′}$/n, where X = [ X$_{1}$, ..., X$_{n}$ ]. Therefore, we consider tests based on λ . If σ 2 is unspecified, the model is invariant with respect to the unitary transfor- mations and multiplications by non-zero scalars, and the maximal invariant is the vector of normalized eigenvalues µ = ( µ$_{1}$, ..., µ$_{m}$$_{−}$$_{1}$ ) , where µ$_{j}$ = λ$_{j}$/ ( λ$_{1}$ + ... + λ$_{p}$ ). Hence, we consider tests based on µ . Note that the distribution of µ does not de- pend on σ $^{2}$, whereas if σ 2 is specified, we can always normalize λ dividing it by σ $^{2}$. Therefore, in what follows, we will assume without loss of generality that σ 2 = 1. Let h = ( h$_{1}$, ..., h$_{r}$ ), and let us denote the joint density of λ$_{1}$, ..., λ$_{m}$ as p$_{λ}$ ( x ; h ) , x = ( x$_{1}$, ..., x$_{m}$ ) ∈ ( R $^{+}$) m and that of µ$_{1}$, ..., µ$_{m}$$_{−}$$_{1}$ as p$_{µ}$ ( y ; h ) , y = ( y$_{1}$, ..., y$_{m}$$_{−}$$_{1}$ ) ∈ 7 2 Likelihood ratios