[Paper Review] Quasi-compactness and absolutely continuous kernels, applications to Markov chains

Hubert Hennion|ArXiv.org|Jun 27, 2006

Mathematical Dynamics and Fractals7 references5 citations

TL;DR

This paper establishes a connection between the essential spectral radius of a bounded positive kernel and its approximation by absolutely continuous kernels, introducing a generalized Doeblin-type condition. It provides a formula for the essential spectral radius and characterizes quasi-compactness of Markov kernels on weighted Banach spaces without requiring irreducibility or aperiodicity, extending classical ergodic and limit theorems to broader settings.

ABSTRACT

We show how the essential spectral radius of a bounded positive kernel, acting on bounded functions, is linked to its lower approximation by certain absolutely continuous kernels. The standart Doeblin's condition can be interpreted in this context, and, when suitably reformulated, it leads to a formula for the essential spectral radius. This results may be used to characterize the Markov kernels having a quasi-compact action on a space of measurable functions bounded with respect to some test function, when no irreducibilty and aperiodicity are assumed.

Motivation & Objective

• To characterize the quasi-compactness of Markov kernels acting on weighted Banach spaces of bounded measurable functions.
• To derive a formula for the essential spectral radius of bounded positive kernels using approximation by absolutely continuous kernels.
• To generalize classical ergodic and limit theorems for Markov chains beyond the standard irreducibility and aperiodicity assumptions.
• To extend the applicability of spectral methods in Markov chain theory via conjugate kernels and Fourier-Laplace transforms.

Proposed method

• Uses Nussbaum's formula for the essential spectral radius as a key analytical tool.
• Introduces a class $\mathcal{K}^*$ of bounded positive absolutely continuous kernels satisfying a uniform integrability condition equivalent to weak compactness on bounded complex measures.
• Applies the Lebesgue-Nikodym decomposition to represent kernels as a sum of an absolutely continuous part and a singular part, enabling approximation.
• Reformulates Doeblin's condition in terms of lower approximation by elements of $\mathcal{K}^*$, leading to a spectral radius formula.
• Analyzes the action of kernels on weighted spaces $\mathcal{B}_w$ via conjugate kernels and adjoint operators.
• Applies results to Fourier and Laplace kernels associated with additive functionals to study limit theorems and large deviations.

Experimental results

Research questions

• RQ1 How can the essential spectral radius of a bounded positive kernel be characterized through approximation by absolutely continuous kernels?
• RQ2 In what way does a generalized Doeblin condition relate to the essential spectral radius and quasi-compactness of a kernel?
• RQ3 Can quasi-compactness of a Markov kernel on a weighted space $\mathcal{B}_w$ be characterized without assuming irreducibility or aperiodicity?
• RQ4 How do conjugate kernels and adjoint operators help in estimating the essential spectral radius in weighted $L^\infty$-type spaces?
• RQ5 What role do Fourier and Laplace kernels play in extending limit theorems to non-irreducible Markov chains?

Key findings

• The essential spectral radius $r_e(Q)$ of a bounded positive kernel $Q$ equals the infimum of $\rho > 0$ such that $Q$ can be approximated in a specific sense by kernels from the class $\mathcal{K}^*$.
• A generalized Doeblin condition, reformulated via approximation by absolutely continuous kernels, provides a formula for $r_e(Q)$.
• Quasi-compactness of a Markov kernel $P$ on $\mathcal{B}_w$ is characterized by the existence of a dominating absolutely continuous kernel satisfying uniform integrability and approximation conditions.
• The essential spectral radius of the conjugate kernel $Q'$ satisfies $r_e(Q') \leq r_e(Q)$, and duality arguments yield $r_e(Q) \leq r_e(Q')$, implying $r_e(Q) = r_e(Q')$.
• The results extend strong and uniform ergodic theorems to non-irreducible and non-aperiodic Markov chains via spectral analysis on weighted spaces.
• The framework supports the study of limit theorems and large deviations through Fourier and Laplace kernels, even when standard assumptions fail.