The modeling and analysis of an LRU cache is extremely challenging as exact results for the main performance metrics (e.g. hit rate) are either lacking or cannot be used because of their high computational complexity for large caches. Recently a TTL-based approximation has been developed for requests described by various workload models and numerically demonstrated to be accurate. The theory for such an approximation, however, is not yet fully developed. In this paper we provide theoretical justification for the approximation in the case where distinct contents are described by independent stationary and ergodic processes. We show that this approximation is exact as the cache size and the number of contents go to infinity. This extends earlier results for the independent reference model. Moreover, we establish results not only for the aggregate cache hit probability but also for every individual content. Last, we obtain bounds on the rate of convergence.