Renewed interest in caching systems stems from their wide-spread use for reducing the document download latency over the Internet. Since caches are usually organized in a hierarchical manner, it is important to study the performance properties of tandem caches. The first step in understanding this problem is to characterize the miss stream from one single cache since it represents the input to the next level cache. In this regard, we discover that the miss stream from one single cache is approximated well by the superposition of a number of asymptotically independent renewal processes. Interestingly, when this weakly correlated miss sequence is fed into another cache, this barely observable correlation can lead to measurably different caching performance when compared to the independent reference model. This result is likely to enable the development of a rigorous analysis of the tandem cache performance.