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

Let uₙ 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 uₙ by extending a method of Mehta and Gurland to the complex case. The asymptotic behavior of E|uₙ|ᵏ as n → ∞ 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.