Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11009_2025_10137_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="131" /> </InlineMediaObject> <EquationSource Format="TEX">$X=\sum _{k=1}^\infty X_k \beta ^{-k}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> <msub> <mi>X</mi> <mi>k</mi> </msub> <msup> <mi>β</mi> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> be the (greedy) base-<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11009_2025_10137_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> expansion of a continuous random variable X on the unit interval where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11009_2025_10137_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> is the positive solution to <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11009_2025_10137_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="174" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta ^n = 1 + \beta + \cdots + \beta ^{n-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>β</mi> <mi>n</mi> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>β</mi> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>β</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> for an integer <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11009_2025_10137_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\geqslant 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> (i.e., <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11009_2025_10137_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> is a generalization of the golden mean corresponding to <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11009_2025_10137_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>). We study the asymptotic distribution and convergence rate of the scaled remainder <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11009_2025_10137_Article_IEq8.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="113" /> </InlineMediaObject> <EquationSource Format="TEX">$\sum _{k=1}^\infty X_{m+k} \beta ^{-k}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> <msub> <mi>X</mi> <mrow> <mi>m</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> <msup> <mi>β</mi> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> when m tends to infinity.

The Asymptotic Distribution of the Scaled Remainder for Pseudo Golden Ratio Expansions of a Continuous Random Variable

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2025年1月30日

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Ira W. Herbst,
Jesper Møller,
Anne Marie Svane

摘要

Let $X=\sum _{k=1}^\infty X_k \beta ^{-k}$ $X = \sum_{k = 1}^{\infty} X_{k} β^{- k}$ be the (greedy) base- $\beta $ $β$ expansion of a continuous random variable X on the unit interval where $\beta $ $β$ is the positive solution to $\beta ^n = 1 + \beta + \cdots + \beta ^{n-1}$ $β^{n} = 1 + β + \dots + β^{n - 1}$ for an integer $n\geqslant 2$ $n ⩾ 2$ (i.e., $\beta $ $β$ is a generalization of the golden mean corresponding to $n=2$ $n = 2$ ). We study the asymptotic distribution and convergence rate of the scaled remainder $\sum _{k=1}^\infty X_{m+k} \beta ^{-k}$ $\sum_{k = 1}^{\infty} X_{m + k} β^{- k}$ when m tends to infinity.