The sampling distribution of the serial correlation coefficient

A methodology for the derivation of the cumulants of the lag-k serial covariance is proposed for the case of a Gaussian white-noise process. Explicit representations of the first five moments of the serial correlation coefficient are then given. Alternate representations which are expressed in terms of trigonometric functions are also derived. The first four moments are used to approximate the density of the serial correlation by means of Pearson curves. A computable representation of the exact distribution function is also given. The exact, the approximate and the simulated distributions are compared in a numerical example.