[proofplan]
We apply the finite-rank covariance form of the [Baik-Ben Arous-Peche Phase Transition for Spiked Covariance Matrices](/theorems/4071). That theorem applies to the sample covariance matrices $\hat{\Sigma}_p$ in the stated Gaussian model and identifies each separated sample-eigenvalue cluster produced by a population spike $1+\theta$ with the deterministic location
\begin{align*}
\rho(\theta)=(1+\theta)\left(1+\frac{\gamma}{\theta}\right)
\end{align*}
when $\theta>\sqrt{\gamma}$. It also states that spikes at or below the BBP threshold do not produce eigenvalues separated by an order-one distance from the Marchenko-Pastur bulk, whose upper edge is $(1+\sqrt{\gamma})^2$. We then use the monotonicity of $\rho$ on $(\sqrt{\gamma},\infty)$ to match the ordered empirical outlier clusters with the ordered list $\theta_{j_1},\dots,\theta_{j_m}$, counting repeated spike strengths with their multiplicities.
[/proofplan]
[step:Apply the finite-rank BBP theorem to each supercritical spike]
Let $\hat{\Sigma}_p$ denote the sample covariance matrix defined in the theorem statement. Define the outlier-location map $\rho:(\sqrt{\gamma},\infty)\to\mathbb{R}$ by
\begin{align*}
\rho(\theta)=(1+\theta)\left(1+\frac{\gamma}{\theta}\right).
\end{align*}
The hypotheses of the finite-rank covariance form of the [Baik-Ben Arous-Peche Phase Transition for Spiked Covariance Matrices](/theorems/4071) are satisfied: the rank $r$ of the perturbation is fixed, $p/n\to\gamma\in(0,\infty)$, the spike strengths are fixed, and the whitened noise matrix $X_p$ has independent standard Gaussian entries. The Marchenko-Pastur bulk has upper edge $(1+\sqrt{\gamma})^2$, with support $[(1-\sqrt{\gamma})^2,(1+\sqrt{\gamma})^2]$ for the absolutely continuous part, using the usual atom-at-zero interpretation when $\gamma>1$. Therefore, for every spike strength $\theta>\sqrt{\gamma}$ with multiplicity $a$, there are exactly $a$ sample eigenvalues of $\hat{\Sigma}_p$ converging in probability to $\rho(\theta)$, and these eigenvalues are separated by an order-one distance from the Marchenko-Pastur bulk edge.
[guided]
The main input is the finite-rank covariance form of the [Baik-Ben Arous-Peche Phase Transition for Spiked Covariance Matrices](/theorems/4071). We first check that it applies in the present setting. The theorem requires a fixed-rank perturbation; the statement assumes that the number $r$ of spikes is fixed. It requires a proportional asymptotic regime; the statement gives $p/n\to\gamma\in(0,\infty)$. It requires fixed population spike strengths; the statement assumes that $\theta_1,\dots,\theta_r$ do not vary with $p$. It also requires the null noise after whitening to satisfy the Gaussian input hypothesis; this is exactly the assumption that $X_p$ has independent standard Gaussian entries in the representation of $\hat{\Sigma}_p$.
For a spike strength $\theta>\sqrt{\gamma}$, define the map $\rho:(\sqrt{\gamma},\infty)\to\mathbb{R}$ by
\begin{align*}
\rho(\theta)=(1+\theta)\left(1+\frac{\gamma}{\theta}\right).
\end{align*}
The finite-rank BBP theorem then states that if the population spike $1+\theta$ has multiplicity $a$, the sample covariance matrix $\hat{\Sigma}_p$ has exactly $a$ eigenvalues outside the Marchenko-Pastur bulk whose common limit in probability is $\rho(\theta)$. The relevant bulk edge is the upper endpoint $(1+\sqrt{\gamma})^2$ of the Marchenko-Pastur support, whose absolutely continuous support is $[(1-\sqrt{\gamma})^2,(1+\sqrt{\gamma})^2]$ with the standard atom-at-zero convention when $\gamma>1$. Thus the BBP theorem supplies both the number of separated eigenvalues and their deterministic limit for each supercritical spike cluster.
[/guided]
[/step]
[step:Order the BBP clusters by the deterministic location map]
For $\theta>\sqrt{\gamma}$, the derivative of $\rho$ is
\begin{align*}
\rho'(\theta)=1-\frac{\gamma}{\theta^2}>0.
\end{align*}
Thus $\rho$ is strictly increasing on $(\sqrt{\gamma},\infty)$. Since the list $\theta_{j_1},\dots,\theta_{j_m}$ is ordered so that the corresponding values $\rho(\theta_{j_k})$ are nonincreasing, the BBP clusters produced in the previous step occur in exactly the same nonincreasing order as the empirical outliers $\lambda_{\mathrm{out},k}(\hat{\Sigma}_p)$, with repeated spike strengths counted by multiplicity. Hence, for each $1\le k\le m$,
\begin{align*}
\lambda_{\mathrm{out},k}(\hat{\Sigma}_p) \to \rho(\theta_{j_k})=(1+\theta_{j_k})\left(1+\frac{\gamma}{\theta_{j_k}}\right)
\end{align*}
in probability.
[/step]
[step:Use the BBP threshold to exclude separated subcritical outliers]
The finite-rank covariance form of the [Baik-Ben Arous-Peche Phase Transition for Spiked Covariance Matrices](/theorems/4071) also gives the threshold statement: if $\theta\le\sqrt{\gamma}$, then the eigenvalues associated with that spike do not converge to a limit strictly outside the Marchenko-Pastur bulk. Equivalently, such spikes do not create order-one separated outlier eigenvalues. This proves the asserted non-separation of all spikes with $\theta_j\le\sqrt{\gamma}$ and completes the proof.
[/step]
