<p>We prove Berry–Esseen theorems for sums <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ S_n=\sum _{j=0}^{n-1}f_j\circ T_{j-1}\circ \cdots \circ T_1\circ T_0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>f</mi> <mi>j</mi> </msub> <mo>∘</mo> <msub> <mi>T</mi> <mrow> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>∘</mo> <mo>⋯</mo> <mo>∘</mo> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>∘</mo> <msub> <mi>T</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f_j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation> are functions with uniformly bounded “variation”and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$T_j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation> is a sequence of expanding maps. Using symbolic representations similar result follow for maps <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$T_j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation> in a small <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> neighborhood of an Axiom A map and Hölder continuous functions <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f_j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation>. All of our results are already new for a single map <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$T_j=T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mi>j</mi> </msub> <mo>=</mo> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation> and a sequence of different functions <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$(f_j)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mi>j</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

Berry Esseen theorems for sequences of expanding maps

期刊论文
发表日期：

2025年3月11日

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Dmitry Dolgopyat,
Yeor Hafouta

摘要

We prove Berry–Esseen theorems for sums $ S_n=\sum _{j=0}^{n-1}f_j\circ T_{j-1}\circ \cdots \circ T_1\circ T_0$ $S_{n} = \sum_{j = 0}^{n - 1} f_{j} \circ T_{j - 1} \circ \dots \circ T_{1} \circ T_{0}$ where $f_j$ $f_{j}$ are functions with uniformly bounded “variation”and $T_j$ $T_{j}$ is a sequence of expanding maps. Using symbolic representations similar result follow for maps $T_j$ $T_{j}$ in a small $C^1$ $C^{1}$ neighborhood of an Axiom A map and Hölder continuous functions $f_j$ $f_{j}$ . All of our results are already new for a single map $T_j=T$ $T_{j} = T$ and a sequence of different functions $(f_j)$ $(f_{j})$ .