Paper: arXiv 2601.04900

Authors: Jean-Gabriel Attali

Abstract

We identify the measurable absorbing obstruction to uniqueness of invariant probability measures for a Markov kernel. Ordinary absorbing decompositions obstruct global irreducibility and recurrence, but not necessarily uniqueness: an absorbing component may have full mass for no invariant probability. We prove that a Markov kernel has more than one invariant probability if and only if it admits a visible absorbing decomposition, namely two disjoint absorbing sets, each having full mass for an invariant probability. The proof uses only the Jordan decomposition of the difference of two invariant probabilities.

Complexity vs Empirical Score

  • Math Complexity: 6.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a clean, rigorous theoretical result characterizing the exact measurable obstruction to the uniqueness of invariant probability measures for Markov kernels. While the measure-theoretic proofs are elegant and clear, the paper is purely theoretical with no empirical evaluation or quantitative financial backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Identify obstruction to uniqueness of invariant probabilities for Markov kernels] --> B[Input: Markov kernel with multiple invariant probability measures]
    B --> C[Method: Apply Jordan decomposition to difference of two invariant probabilities]
    C --> D[Analysis: Distinguish ordinary vs. visible absorbing decompositions]
    D --> E[Outcome: Proof that >1 invariant probability iff a visible absorbing decomposition exists]

Cited References (15)

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  2. Ergodicity of a certain class of non-Feller models: applications to ARCH and Markov switching models (2004)
  3. An Introduction to Ergodic Theory (1982)
  4. Ergodic Theory (1983)
  5. Yet another look at H arris’ ergodic theorem for M arkov chains (2011)
  6. Subgeometric rates of convergence of $f$ -ergodic strong M arkov processes (2009)
  7. Random Iterative Models (1997)
  8. Modeling stock returns by a generalized autoregressive conditional heteroskedasticity model (2000)
and 7 more
  1. Ergodic Theory (1983)
  2. An Introduction to Ergodic Theory (1982)
  3. Méthodes récursives aléatoires (1990)
  4. Méthode de stabilit 'e pour des cha ^\i nes de Markov non fellériennes (1999)
  5. The strong law of large numbers for a class of Markov chains (1960)
  6. Strong ergodicity and the strong law of large numbers (1961)
  7. Foundations of Modern Probability (2002)

0 of 15 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.