When a Markov kernel P satisfies a minorization condition and nested modulated drift conditions, Jarner and Roberts provided an asymptotic polynomial convergence rate in weighted total variation norm of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1416_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$P^n(x,\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>P</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to the invariant probability measure <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1416_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation> of P. In connection with this polynomial asymptotic, we propose explicit and simple estimates on series of such weighted total variation norms, from which an estimate for the total variation norm of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1416_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="89" /> </InlineMediaObject> <EquationSource Format="TEX">$P^n(x,\cdot )-\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>P</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> is deduced. The proofs are self-contained and based on the residual kernel and the Nummelin-type representation of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1416_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation>. No coupling technique is used.

Polynomial Convergence Rates for Markov Kernels Under Nested Modulated Drift Conditions

期刊论文
发表日期：

2025年5月10日

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Loïc Hervé,
James Ledoux

摘要

When a Markov kernel P satisfies a minorization condition and nested modulated drift conditions, Jarner and Roberts provided an asymptotic polynomial convergence rate in weighted total variation norm of $P^n(x,\cdot )$ $P^{n} (x, \cdot)$ to the invariant probability measure $\pi $ $π$ of P. In connection with this polynomial asymptotic, we propose explicit and simple estimates on series of such weighted total variation norms, from which an estimate for the total variation norm of $P^n(x,\cdot )-\pi $ $P^{n} (x, \cdot) - π$ is deduced. The proofs are self-contained and based on the residual kernel and the Nummelin-type representation of $\pi $ $π$ . No coupling technique is used.