Convergence of Krasulina Scheme

Principal component analysis (PCA) is one of the most commonly used statistical procedures with a wide range of applications. Consider the points $X_1, X_2,..., X_n$ are vectors drawn i.i.d. from a distribution with mean zero and covariance $\Sigma$, where $\Sigma$ is unknown. Let $A_n = X_nX_n^T$, then $E[A_n] = \Sigma$. This paper consider the problem of finding the least eigenvalue and eigenvector of matrix $\Sigma$. A classical such estimator are due to Krasulina\cite{krasulina_method_1969}. We are going to state the convergence proof of Krasulina for the least eigenvalue and corresponding eigenvector, and then find their convergence rate.