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On the strong stability of ergodic iterations

László Györfi, Attila Lovas, Miklós Rásonyi

TL;DR

This work revisits processes generated by iterated random functions driven by a stationary and ergodic sequence and shows the strong stability of iterations under some mild conditions on the corresponding recursive map.

Abstract

We revisit processes generated by iterated random functions driven by a stationary and ergodic sequence. Such a process is called strongly stable if a random initialization exists, for which the process is stationary and ergodic, and for any other initialization, the difference between the two processes converges to zero almost surely. Under some mild conditions on the corresponding recursive map, without any condition on the driving sequence, we show the strong stability of iterations. Several applications are surveyed such as generalized autoregression and queuing. Furthermore, new results are deduced for Langevin-type iterations with dependent noise and for multitype branching processes.

On the strong stability of ergodic iterations

TL;DR

This work revisits processes generated by iterated random functions driven by a stationary and ergodic sequence and shows the strong stability of iterations under some mild conditions on the corresponding recursive map.

Abstract

We revisit processes generated by iterated random functions driven by a stationary and ergodic sequence. Such a process is called strongly stable if a random initialization exists, for which the process is stationary and ergodic, and for any other initialization, the difference between the two processes converges to zero almost surely. Under some mild conditions on the corresponding recursive map, without any condition on the driving sequence, we show the strong stability of iterations. Several applications are surveyed such as generalized autoregression and queuing. Furthermore, new results are deduced for Langevin-type iterations with dependent noise and for multitype branching processes.
Paper Structure (6 sections, 7 theorems, 81 equations)

This paper contains 6 sections, 7 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Iterated ergodic function systems
  3. Generalized autoregression
  4. Lindley process
  5. The Langevin iteration
  6. Generalized multi-type Galton--Watson process

Key Result

Theorem 1

Assume that $\{Z_i\}_{-\infty}^{\infty}$ is a stationary and ergodic sequence. Suppose that Then the class $\{X_n(v), v\in{\mathbb R}^d\}$ is strongly stable.

Theorems & Definitions (23)

  • Definition 1
  • Theorem 1
  • proof : Proof of Theorem \ref{['gen']}.
  • Proposition 1
  • proof
  • Definition 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 2
  • ...and 13 more