Abstract
We prove that the sum of t boolean-valued random variables sampled by a random walk on a regular expander converges in total variation distance to a discrete normal distribution at a rate of O(λ/t1/2−o(1)), where λ is the second largest eigenvalue of the random walk matrix in absolute value. To the best of our knowledge, among known Berry-Esseen bounds for Markov chains, our result is the first to show convergence in total variation distance, and is also the first to incorporate a linear dependence on expansion λ. In contrast, prior Markov chain Berry-Esseen bounds showed a convergence rate of O(1/√t) in weaker metrics such as Kolmogorov distance.
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