Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A convenient characterisation of convergent upper transition operators

Jasper De Bock, Alexander Erreygers, Floris Persiau

TL;DR

This paper advances the understanding of the limit behavior of imprecise Markov chains by studying the convergence of upper transition operators $\overline{T}$ on finite state spaces. It introduces a general, practically verifiable sufficient condition for convergence based on upper accessibility and lower reachability, and shows that this condition is necessary in the cases where maximal communication classes are absorbing or when $\overline{T}$ is finitely generated. The authors develop a graph-theoretic and restriction-based framework to decompose the state space into maximal and transient classes, derive ergodicity criteria, and provide an algorithmic procedure for the finitely generated case to certify convergence. They also present a counterexample illustrating that convergence can occur even when the restricted dynamics are not convergent, and discuss paths for extending results beyond the finitely generated setting and connections to Akian’s theorem.

Abstract

Motivated by its connection to the limit behaviour of imprecise Markov chains, we introduce and study the so-called convergence of upper transition operators: the condition that for any function, the orbit resulting from iterated application of this operator converges. In contrast, the existing notion of `ergodicity' requires convergence of the orbit to a constant. We derive a very general (and practically verifiable) sufficient condition for convergence in terms of accessibility and lower reachability, and prove that this sufficient condition is also necessary whenever (i) all transient states are absorbed or (ii) the upper transition operator is finitely generated.

A convenient characterisation of convergent upper transition operators

TL;DR

This paper advances the understanding of the limit behavior of imprecise Markov chains by studying the convergence of upper transition operators on finite state spaces. It introduces a general, practically verifiable sufficient condition for convergence based on upper accessibility and lower reachability, and shows that this condition is necessary in the cases where maximal communication classes are absorbing or when is finitely generated. The authors develop a graph-theoretic and restriction-based framework to decompose the state space into maximal and transient classes, derive ergodicity criteria, and provide an algorithmic procedure for the finitely generated case to certify convergence. They also present a counterexample illustrating that convergence can occur even when the restricted dynamics are not convergent, and discuss paths for extending results beyond the finitely generated setting and connections to Akian’s theorem.

Abstract

Motivated by its connection to the limit behaviour of imprecise Markov chains, we introduce and study the so-called convergence of upper transition operators: the condition that for any function, the orbit resulting from iterated application of this operator converges. In contrast, the existing notion of `ergodicity' requires convergence of the orbit to a constant. We derive a very general (and practically verifiable) sufficient condition for convergence in terms of accessibility and lower reachability, and prove that this sufficient condition is also necessary whenever (i) all transient states are absorbed or (ii) the upper transition operator is finitely generated.

Paper Structure

This paper contains 17 sections, 25 theorems, 69 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Upper transition operators
  3. Convergence and ergodicity
  4. Convenient properties of upper transition operators
  5. Characterising ergodicity
  6. The upper accessibility graph
  7. Lower reachability and absorption
  8. A necessary and sufficient condition for ergodicity
  9. Conditions for convergence
  10. A single communication class
  11. Restriction
  12. Maximal classes are absorbing
  13. Maximal classes are not absorbing
  14. The finitely generated case
  15. Wrapping things up
  16. ...and 2 more sections

Key Result

lemma 1

Consider an upper transition operator $\overline{T}{}$ with closed class $C$. Let $(C_n)_{n\in\mathbb{Z}_{\geq0}}$ be the non-decreasing sequence given by $C_0\coloneq C$ and, for all $n\in\mathbb{Z}_{\geq0}$, by Then after $k\leq\vert\mathcal{X}\setminus C\vert$ steps, we reach $C_k=C_{k+1}$, and $C_k$ is the set of states from which $C$ is lower reachable.

Figures (5)

  • Figure 1: $\overline{\mathcal{G}}(\overline{T}{})$ for \ref{['example:running example first part']}.
  • Figure 2: Venn diagram of the state space $\mathcal{X}$
  • Figure 3: $\overline{\mathcal{G}}({\overline{T}{}_{\mathcal{X}_{\mathrm{t}\triangledown}}})$ for \ref{['example:running example eerste keer knippen']}.
  • Figure 4: Modified Venn diagram of the state space $\mathcal{X}$
  • Figure 5:

Theorems & Definitions (47)

  • definition 1
  • definition 2
  • lemma 1
  • proof
  • lemma 2
  • proof
  • proposition 1
  • proposition 2
  • lemma 3
  • proof
  • ...and 37 more