The estimation of the correlation coefficient of bivariate data under dependence: Convergence analysis

Let {Xi,Yi}{Xi,Yi} be jointly distributed second-order random variables with correlation coefficient rr. The estimation of rr from the observations {Xi,Yi}i=1n is a classical problem which has been examined under the assumption of an i.i.d. setting. In this paper we examine the statistical properties of the correlation coefficient estimate when the process {Xi,Yi}{Xi,Yi} is dependent, constituting either a strongly mixing process or asymptotically uncorrelated. We establish convergence in probability (with rates) as well as asymptotic normality for the estimation error and present an explicit expression for the asymptotic variance.