Some exact sampling distributions for variogram estimators

For equally spaced observations from a one-dimensional, stationary, Gaussian random function, the characteristic function of the usual variogram estimator for a fixed lag k is derived. Because the characteristic function and the probability density function form a Fourier integral pair, it is possible to tabulate the sampling distribution of a function of a using either analytic or numerical methods. An example of one such tabulation is given for an underlying model that is simple transitive.