On the Correlation Coefficient of a Bivariate, Equal Variance, Complex Gaussian Sample

Abstract

Let $u_n$ denote the sample correlation coefficient for $n$ observations from a bivariate, equal variance, complex Gaussian distribution. In this note we derive the exact distribution of $u_n$ by extending a method of Mehta and Gurland to the complex case. The asymptotic behavior of $E|u_n|^k$ as $n \rightarrow \infty$ is determined via the method of steepest descent. Applicability of the results to the analysis of certain estimators of spectral parameters of stationary time series is discussed.