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<p>We adapt Stein’s method of diffusion approximations, developed by Barbour, to the study of chaotic dynamical systems. We establish an error bound in the functional central limit theorem with respect to an integral probability metric of smooth test functions under a functional correlation decay bound. For systems with a sufficiently fast polynomial rate of correlation decay, the error bound is of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(O(N^{-1/2})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>N</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, under an additional condition on the linear growth of variance. Applications include a family of interval maps with neutral fixed points and unbounded derivatives, and two-dimensional dispersing Sinai billiards.</p>

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Error Bounds in a Smooth Metric for Brownian Approximation of Dynamical Systems via Stein’s Method

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年11月19日

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  • Juho Leppänen,
  • Yuto Nakajima,
  • Yushi Nakano

摘要

We adapt Stein’s method of diffusion approximations, developed by Barbour, to the study of chaotic dynamical systems. We establish an error bound in the functional central limit theorem with respect to an integral probability metric of smooth test functions under a functional correlation decay bound. For systems with a sufficiently fast polynomial rate of correlation decay, the error bound is of order \(O(N^{-1/2})\) O ( N - 1 / 2 ) , under an additional condition on the linear growth of variance. Applications include a family of interval maps with neutral fixed points and unbounded derivatives, and two-dimensional dispersing Sinai billiards.