Abstract
Let \(z (n) \in {\mathbb {Z}}^d, n = 0,1,\ldots ,\) be a centered random walk stopped the first time it exits \(\Omega \subset {\mathbb {R}}^d\) some “upper half space” with Lipschitz boundary (cf. Sect. 1.2.1) and let \(p_n^{\mu } (x,y)\) be the transition probability (\(\mu \) as in Sect. 1.2.2). Let \(p_t (x, y) \ x, y \in \Omega , t > 0\), be the transition density of the normalized Brownian motion stopped upon exit from \(\Omega \). I give an optimal estimate of \(p_t^{\mu } - p_t\) that is the generalization of the Edgeworth expansion of the Central Limit Theorem to Lipschitz domains.
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Varopoulos, N.T. The central limit theorem in Lipschitz domains. Boll Unione Mat Ital 7, 103–156 (2014). https://doi.org/10.1007/s40574-014-0005-x
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DOI: https://doi.org/10.1007/s40574-014-0005-x