  In this paper, the key objects of interest are the sequential covariance
matrices $\mathbf{S}_{n,t}$ and their largest eigenvalues. Here, the matrix
$\mathbf{S}_{n,t}$ is computed as the empirical covariance associated with
observations $\{\mathbf{x}_1,\ldots,\mathbf{x}_{ \lfloor nt \rfloor } \}$, for
$t\in [0,1]$. The observations $\mathbf{x}_1,\ldots,\mathbf{x}_n$ are assumed
to be i.i.d. $p$-dimensional vectors with zero mean, and a covariance matrix
that is a fixed-rank perturbation of the identity matrix. Treating $\{
\mathbf{S}_{n,t}\}_{t \in [0,1]}$ as a matrix-valued stochastic process indexed
by $t$, we study the behavior of the largest eigenvalues of $\mathbf{S}_{n,t}$,
as $t$ varies, with $n$ and $p$ increasing simultaneously, so that $p/n \to y
\in (0,1)$. As a key contribution of this work, we establish the weak
convergence of the stochastic process corresponding to the sample spiked
eigenvalues, if their population counterparts exceed the critical
phase-transition threshold. Our analysis of the limiting process is fully
comprehensive revealing, in general, non-Gaussian limiting processes.
  As an application, we consider a class of change-point problems, where the
interest is in detecting structural breaks in the covariance caused by a change
in magnitude of the spiked eigenvalues. For this purpose, we propose two
different maximal statistics corresponding to centered spiked eigenvalues of
the sequential covariances. We show the existence of limiting null
distributions for these statistics, and prove consistency of the test under
fixed alternatives. Moreover, we compare the behavior of the proposed tests
through a simulation study.
