  Since its inception in 1982, Oja's algorithm has become an established method
for streaming principle component analysis (PCA). We study the problem of
streaming PCA, where the data-points are sampled from an irreducible,
aperiodic, and reversible Markov chain. Our goal is to estimate the top
eigenvector of the unknown covariance matrix of the stationary distribution.
This setting has implications in scenarios where data can solely be sampled
from a Markov Chain Monte Carlo (MCMC) type algorithm, and the objective is to
perform inference on parameters of the stationary distribution. Most
convergence guarantees for Oja's algorithm in the literature assume that the
data-points are sampled IID. For data streams with Markovian dependence, one
typically downsamples the data to get a "nearly" independent data stream. In
this paper, we obtain the first sharp rate for Oja's algorithm on the entire
data, where we remove the logarithmic dependence on the sample size, $n$,
resulting from throwing data away in downsampling strategies.
