<p>We consider the products <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1434_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="109" /> </InlineMediaObject> <EquationSource Format="TEX">$G_n = A_n \cdots A_1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mi>n</mi> </msub> <mo>=</mo> <msub> <mi>A</mi> <mi>n</mi> </msub> <mo>⋯</mo> <msub> <mi>A</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> of independent and identically distributed nonnegative <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1434_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$d \times d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>×</mo> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation> matrices <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1434_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$(A_i)_{i \geqslant 1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>i</mi> <mo>⩾</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>. For any starting point <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1434_Article_IEq4.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$x \in {\mathbb {R}}_+^d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <msubsup> <mi mathvariant="double-struck">R</mi> <mo>+</mo> <mi>d</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation> with unit norm, we establish the convergence to a stable law for the norm cocycle <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1434_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">$\log | G_nx |$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo>log</mo> <mo stretchy="false">|</mo> </mrow> <msub> <mi>G</mi> <mi>n</mi> </msub> <mrow> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, jointly with its direction <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1434_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="148" /> </InlineMediaObject> <EquationSource Format="TEX">$G_n \cdot x = G_n x / | G_n x |$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mi>n</mi> </msub> <mo>·</mo> <mi>x</mi> <mo>=</mo> <msub> <mi>G</mi> <mi>n</mi> </msub> <mi>x</mi> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">|</mo> <msub> <mi>G</mi> <mi>n</mi> </msub> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We also prove a local limit theorem for the couple <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1434_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="127" /> </InlineMediaObject> <EquationSource Format="TEX">$ (\log |G_nx|, G_n \cdot x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mo stretchy="false">|</mo> <msub> <mi>G</mi> <mi>n</mi> </msub> <mi>x</mi> <mo stretchy="false">|</mo> <mo>,</mo> <msub> <mi>G</mi> <mi>n</mi> </msub> <mo>·</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and find the exact rate of its convergence.</p>

Convergence to Stable Laws and a Local Limit Theorem for Products of Positive Random Matrices

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2025年9月2日

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Jianzhang Mei,
Quansheng Liu

摘要

We consider the products $G_n = A_n \cdots A_1$ $G_{n} = A_{n} \dots A_{1}$ of independent and identically distributed nonnegative $d \times d$ $d \times d$ matrices $(A_i)_{i \geqslant 1}$ ${(A_{i})}_{i ⩾ 1}$ . For any starting point $x \in {\mathbb {R}}_+^d$ $x \in R_{+}^{d}$ with unit norm, we establish the convergence to a stable law for the norm cocycle $\log | G_nx |$ $log | G_{n} x |$ , jointly with its direction $G_n \cdot x = G_n x / | G_n x |$ $G_{n} \cdot x = G_{n} x / | G_{n} x |$ . We also prove a local limit theorem for the couple $ (\log |G_nx|, G_n \cdot x)$ $(log | G_{n} x |, G_{n} \cdot x)$ and find the exact rate of its convergence.