A Central Limit Theorem for Non-Overlapping Return Times

Abstract

Define the non-overlapping return time of a block of a random process to be the number of blocks that pass by before the block in question reappears. We prove a central limit theorem based on these return times. This result has applications to entropy estimation, and to the problem of determining if digits have come from an independent, equidistributed sequence. In the case of an equidistributed sequence, we use an argument based on negative association to prove convergence under conditions weaker than those required in the general case.

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“…Maurer [35] proved that log S 1 / converges to the entropy H if the source is IID binary, with a similar result proved for stationary ψ-mixing processes by Abadi and Galves in [1]. Johnson [27] proved a Central Limit Theorem for the average of log S i , and hence consistency of the resulting entropy estimates.…”