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The first order Gaussian auto-regressive process (xₜ) may be defined by the stochastic difference equation equation*1xₜ = xₓ-₁ + uₜ, equation* where the u's are NID (0, 1) and is an unknown parameter. The choice of a statistic as an estimator for depends on the initial conditions imposed on the difference equation (1). The so-called "circular" model is obtained by considering a sample of size N and then assuming that x₍ + ₁ = x₁. An appropriate estimator for in this case is the circular serial correlation coefficient equation*2 r = Nₓ = ₁ ₗ䂻ₗ_ₓ + ₁Nₓ = ₁ x²ₜ (x₍ + ₁ = x₁). equation* Leipnik 1 has derived an approximate density function equation*3 f (t) = (N + 2{2) } (N + 1{2 (12) } (1 - 2t + ²) ^-N/2 (1 - t²) ^ (N - 1) /2equation* for the estimator r. Leipnik also evaluated the first two moments of this distribution. In this paper a formula is obtained which gives E (rᵏ) as a polynomial of degree k in.
John S. White (Sun,) studied this question.