The variance of correlation function estimates

Landy & Szalay (1993) present a new estimator for angular correlation functions, and an expression for the variance of this estimator for the special case of Poisson-distributed points. We extend their results to give the variance of the estimator in the case of correlated point sets and also derive the covariance between estimates at different angular separations. These new covariance formulae are accurate when the number of objects N is > 1, the field is much larger than the angular separations under consideration, and the area-averaged two-point correlation wOMEGA_ is < 1. Although the covariance matrix depends upon the three- and four-point correlation functions, it is simply expressed in terms of the two-point correlation function if the clustering follows a hierarchical form. Use of the Poisson errors greatly misestimates the covariance matrix when N becomes large and leads to underestimates of the uncertainties on the fitted parameters of model correlation functions. The new covariance matrix formula is shown to be accurate to 10% for Monte Carlo data. Fitting of model correlation functions to observations is complicated by the significant cross- correlations between bins. We fit power-law models to the Monte Carlo data as an example.