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On a problem of Ö. Stenflo

Klaudiusz Czudek, Tomasz Szarek

Abstract

We construct an e-chain on a locally compact space with the unique stationary distribution such that the strong law of large numbers does not hold. This answers negatively the question asked by Ö. Stenflo.

On a problem of Ö. Stenflo

Abstract

We construct an e-chain on a locally compact space with the unique stationary distribution such that the strong law of large numbers does not hold. This answers negatively the question asked by Ö. Stenflo.
Paper Structure (3 sections, 3 theorems, 20 equations)

This paper contains 3 sections, 3 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Auxiliary Lemmas
  3. Proof of Theorem \ref{['T']}

Key Result

Theorem 1

Assume that $(Z_n)_{n\ge 0}$ is a Markov chain on $[0, 1)$ associated with the Iterated Function System $(f_1, f_2; \tfrac{1}{2}, \tfrac{1}{2})$. Then $(Z_n)_{n\ge 0}$ is an e--chain admitting a unique invariant measure $\mu_*=\delta_0$. Moreover, for any $x\in (0, 1)$ and $\varphi\in C_c([0, 1))$ s for $\omega$ in some set $\Omega_0$ with $\mathbb P(\Omega_0)>0$.

Theorems & Definitions (6)

  • Theorem 1
  • Remark 2
  • Lemma 3
  • Remark 4
  • Lemma 5
  • Remark 6