process { L ( h ; λ ) ; h ∈ [ 0 , ¯ h ] r } , where h = ( h$_{1}$, ..., h$_{r}$ ) , ¯ h ∈ [0 $_{,}$√$_{c}$$_{) and}$ L ( h ; λ ) is defined as the ratio of the density of λ under H$_{1}$ to that under H$_{0}$ , considered as a λ -measurable random variable. Note that L ( h ; λ ) depends on n and p, while λ is m = min { n, p } -dimensional. An exact formula for L ( h ; λ ) involves the integral ∫ O ( p $_{)}$e tr( AQBQ ′ $^{)}$( dQ ) over the orthogonal group O ( p ), where the p × p matrix A has a deficient rank r . In the single-spiked case ( r = 1), OMH link this integral to the confluent form of the Lauricella function, and use this link to establish a representation of the integral in the form of a contour integral (see Wang (2010) and Mo (2011) for independent different derivations of this contour integral representation for this particular r = 1 case). Then, the Laplace approximation to the contour integral is used to derive the asymptotic behavior of L ( h ; λ ). Onatski (2012) generalizes the contour integral representation to the multi- spiked case ( r > 1). For complex-valued data, such a generalization allows him to extend OMH’s results to the multi-spiked context. Unfortunately, for real- valued data, which we are concerned with in this paper, this generalization is not straightforwardly amenable to the Laplace approximation method. Therefore, in this paper, we consider a totally different approach. For the r = 1 case, Guionnet and Maida (2005) (hereafter GM) use large deviation methods to derive a second- order asymptotic expansion of ∫ O ( p $_{)}$e tr( AQBQ ′ $^{)}$( dQ ) as the non-zero eigenvalues of A diverge to infinity (see their Theorem 3). We extend GM’s second-order expansion to the r > 1 case, and use that extension to derive the asymptotics of L ( h ; λ ). More precisely, we show that, for any ¯ h such that 0 < ¯ h < √$_{c,}$ the sequence of log-likelihood processes { ln L ( h ; λ ) ; h ∈ [0 , ¯ h ] $^{r}$} converges weakly to a Gaussian process { L $_{λ}$( h ); h ∈ [ 0 , ¯ h ] r } under the null hypothesis H$_{0}$ as n, p → c ∞ . The index λ in the notation L $_{λ}$( h ) is used to distinguish the limiting λ -log-likelihood process in the case of specified σ 2 = 1 , from that of the µ -log-likelihood process considered in the case of unspecified σ $^{2}$, which we denote by L $_{µ}$( h ) (see Section 2). The limiting 4