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BerryEsseen type theorem
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Albanese metric Alexopoulos argument assume Asymptotic behavior Berry-Esseen type theorem bipartite graph Cayley graph central limit theorem converges Coulhon covering transformation group define denote discrete group estimates for kn exists a constant exp dx(x exp_1(x exponential map fc=J following hold fundamental domain Gaussian estimates Gaussian upper estimate Gr,Gr group Gr group of polynomial harmonic maps harmonic realization heat kernel Hebisch and Saloff-Coste Hence integer invariant vector field isomorphic Kagome lattice kn(x Kotani and Sunada left invariant vector Lemma Lie algebra limit group limit operator long time asymptotics LP boundedness manifolds Math nilpotent covering graph nilpotent Lie group non-bipartite graph polynomial growth polynomial volume growth proof of Theorem random walks Riemannian manifolds Riesz transform Saloff-Coste 15 Section 2.1 semigroup stratified Lie group sub-Laplacian Sunada 22 symmetric random walk T-equivariant theorem on nilpotent Tohoku University transition operator transition probability weak-(l