Centrifugal pendulum vibration absorbers are used for reducing engine-order torsional vibrations in rotating machines. The most common configuration of these devices utilizes a bifilar suspension in which the absorber mass is suspended by a pair of cylindrical rollers that allow it to move along a prescribed path that is determined by the shape of machined cutouts on the rotor and the absorber mass. Previous studies have considered how to account for the roller inertia in selecting the linear (small amplitude) tuning characteristics of the absorber system. Here, we describe a systematic study of the nonlinear (finite amplitude) aspects of this system and show that there exists an absorber path for which the absorber/roller system maintains the same frequency of free oscillation over all physically possible amplitudes when the rotor spins at a constant rate. This tautochronic path has been well known for the case with zero roller inertia, and herein, for the first time, the corresponding path with rollers is shown to exist, and a method for its construction is presented. In addition, we carry out a perturbation analysis of the steady-state dynamic response of the rotor/absorber/roller system in order to quantify the effects of various approximations commonly used with regard to the roller dynamics. The results show that if one accounts for the rollers in the linear absorber tuning, the nonlinear system response is essentially insensitive to the selection of the nonlinear tuning parameter, so long as it is close to the tautochronic value.