In this paper we focus on the LRU cache where requests for distinct contents are described by independent stationary and ergodic processes. We extend a TTL-based approximation of the cache hit probability rst proposed by Fagin [14] for the independence reference model to this more general workload model. We show that under very general conditions this approximation is exact as the cache size and the number of contents go to innnity. Moreover, we establish this not only for the aggregate cache hit probability as in [14] but also for every individual content. Last, we obtain a rate of convergence.