Linear TD($\lambda$) is one of the fundamental reinforcement learning algorithms for policy evaluation. While the convergence rate has traditionally been determined by assuming linear independence of features, this assumption often fails in many real-world situations. This paper establishes the first $L^2$ convergence rate for Linear TD($\lambda$) operating on arbitrary features without modifying the algorithm or making additional assumptions. This holds true for both discounting and average reward settings. Furthermore, to address the non-uniformity of potential solutions due to arbitrary features, we develop a novel probabilistic approximation that characterizes the convergence rate to a set of solutions rather than a single point.
