The asymptotic behavior of an N-electron ground-state wave function is analyzed, as one electron wanders far from the system. Implications for the one-matrix and pair density are described. The asymptotic behavior currently discussed in the literature, in which the remaining (N-1) electrons relax to their ground state, is generalized to the case where the (N-1)-electron ground state is degenerate. Infinitely long-ranged correlations are reported, in which the selected (N-1)-electron ground state depends upon the direction along which, one electron wandered off. We comet a standard limit for the one matrix. Numerical and analytic studies of accurate correlated wave functions illustrate and support the standard asymptotic behavior for the nondegenerate case and its generalization derived here. We extract the (N-1)-electron density from the correlated N-electron wave function. We also discuss the question how large the separation of one electron must be to realize the limiting behavior.