We establish an almost sure convergence theorem of the stochasticapproximation process of Oja for estimating eigenvectors of theQ-symmetric expectation of a random matrix, under a correlationmodel between the incoming random matrices. This theorem gener-alizes previous theorems and extends them to the case where themetricQis unknown and estimated online in parallel. We apply it tostreaming principal component analysis of a random vectorZ, whena mini-batch of observations ofZis used at each step or all theobservations up to the current step. We deal with the case ofstreaming generalized canonical correlation analysis, with a metricestimated online in parallel.