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Cache replacement policies typically use some form of statistics on past access behavior. As a common limitation, how- ever, the extent of the history being recorded is limited to either just the data in cache or, more recently, a larger but still finite-length window of accesses, because the cost of keeping a long history can easily outweigh its benefit. This paper presents a statistical method to keep track of instruction pointer-based access reuse intervals of arbitrary length and uses this information to identify the Least Ex- pected Use (LEU) blocks for replacement. LEU uses dynamic sampling supported by novel hardware that maintains a state to record arbitrarily long reuse intervals. LEU is evaluated using the Cache Replacement Championship simulator, tested on PolyBench and SPEC, and compared with five policies including a recent technique that approximates optimal caching using a fixed-length history. By maintaining statistics for an arbitrary history, LEU outperforms previous techniques for a broad range of scientific kernels, whose data reuses are longer than those in traces traditionally used in computer architecture studies.