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 $O(\lambda/t^{1/2-o(1)})$, where $\lambda$ 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 $\lambda$. In contrast, prior Markov chain Berry-Esseen bounds showed a convergence rate of $O(1/\sqrt{t})$ in weaker metrics such as Kolmogorov distance. Our result also improves upon prior work in the pseudorandomness literature, which showed that the total variation distance is $O(\lambda)$ when the approximating distribution is taken to be a binomial distribution. We achieve the faster $O(\lambda/t^{1/2-o(1)})$ convergence rate by generalizing the binomial distribution to discrete normals of arbitrary variance. We specifically construct discrete normals using a random walk on an appropriate 2-state Markov chain. Our bound can therefore be viewed as a regularity lemma that reduces the study of arbitrary expanders to a small class of particularly simple expanders.

翻译：我们证明，由正则展开图上的随机游走所抽取的$t$个布尔值随机变量之和，在总变差距离下以$O(\lambda/t^{1/2-o(1)})$的速率收敛于离散正态分布，其中$\lambda$是随机游走矩阵的第二大特征值的绝对值。据我们所知，在已有的马尔可夫链Berry-Esseen界中，我们的结果是首个证明总变差距离收敛性的工作，也是首个将收敛速率与展开度$\lambda$建立线性依赖关系的工作。相比之下，先前的马尔可夫链Berry-Esseen界在Kolmogorov距离等较弱度量下仅给出$O(1/\sqrt{t})$的收敛速率。我们的结果还改进了伪随机性文献中的前期工作，该工作表明当逼近分布取为二项分布时，总变差距离为$O(\lambda)$。我们通过将二项分布推广为任意方差的离散正态分布，实现了更快的$O(\lambda/t^{1/2-o(1)})$收敛速率。具体地，我们利用在恰当的二态马尔可夫链上的随机游走来构造离散正态分布。因此，我们的界可被视作一个正则性引理，它将任意展开图的研究简化为特别简单的一小类展开图。