Distribution of the Serial Correlation Coefficient in a Circularly Correlated Universe

It is desired to find an approximate distribution of simple form for the statistic r = x₁x₂ + ⋯ + xTx₁/x₁² + ⋯ + xT² (r is an estimate of the serial correlation coefficient ρ in a circular universe) in the case that ρ ≠ O in the universe. Such a distribution is obtained by smoothing the joint characteristic function of the numerator and denominator of the expression for r. The first two moments are calculated; from these r is seen to be a consistent estimate of ρ. A graph of this distribution for sample size $T = 20$ and various values of ρ is given. In addition, an approximate distribution for p = x²₁ + ⋯ + x²T is derived which reduces to the exact (χ²-) distribution if ρ = 0. From a formula which yields all moments, it is concluded that, at least up to the degree of approximation attained, $p/T$ is an unbiased and consistent extimate of σ².