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Markov chains, AR linear models, and regular variation

Piotr Dyszewski, Tamara Mika

TL;DR

The paper develops a unified framework for multivariate regular variation of vector- and Banach-space–valued stochastic processes, focusing on time-homogeneous Markov chains and random coefficient linear models. It establishes that, under a gauge-monotonicity condition and other regularity/ergodicity assumptions, the extremes of the process inherit regular variation from the innovations, with explicitly described spectral measures; it further analyzes two regimes for random coefficient models—adaptable and predictable—yielding tail indices $\alpha/2$ and $\alpha$, respectively, along with their spectral structure. A central application to random difference equations demonstrates how a stationary solution can be represented as an infinite series with regularly varying tails, and reveals cone-specific regular variation for extremal events. The results provide concrete tools for modeling and quantifying joint extremes in complex stochastic systems, including financial or network contexts, by linking tail behavior to the underlying Markov dynamics and coefficient structure.

Abstract

We investigate multivariate regular variation in the context of time-homogeneous Markov chains on general vector spaces and in random coefficient linear models. In the first part, we show that the regular variation of the stationary distribution can be derived from that of the innovations, provided that the chain satisfies a certain monotonicity condition with respect to a gauge function. In the second part, we study random linear models with random coefficients defined by an explicit iterative scheme. We prove that the precise structure of the underlying chain affects the form of the associated spectral measure.

Markov chains, AR linear models, and regular variation

TL;DR

The paper develops a unified framework for multivariate regular variation of vector- and Banach-space–valued stochastic processes, focusing on time-homogeneous Markov chains and random coefficient linear models. It establishes that, under a gauge-monotonicity condition and other regularity/ergodicity assumptions, the extremes of the process inherit regular variation from the innovations, with explicitly described spectral measures; it further analyzes two regimes for random coefficient models—adaptable and predictable—yielding tail indices and , respectively, along with their spectral structure. A central application to random difference equations demonstrates how a stationary solution can be represented as an infinite series with regularly varying tails, and reveals cone-specific regular variation for extremal events. The results provide concrete tools for modeling and quantifying joint extremes in complex stochastic systems, including financial or network contexts, by linking tail behavior to the underlying Markov dynamics and coefficient structure.

Abstract

We investigate multivariate regular variation in the context of time-homogeneous Markov chains on general vector spaces and in random coefficient linear models. In the first part, we show that the regular variation of the stationary distribution can be derived from that of the innovations, provided that the chain satisfies a certain monotonicity condition with respect to a gauge function. In the second part, we study random linear models with random coefficients defined by an explicit iterative scheme. We prove that the precise structure of the underlying chain affects the form of the associated spectral measure.
Paper Structure (15 sections, 14 theorems, 188 equations)

This paper contains 15 sections, 14 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Multivariate regular variation
  4. Markov chains
  5. Random coefficient linear models
  6. A general setting
  7. The adaptable case
  8. The predictable case
  9. Random difference equation
  10. Proofs
  11. Multivariate regular variation
  12. Markov chains
  13. Random coefficient linear models
  14. Adaptable case
  15. Predictable case

Key Result

Proposition 2.3

Let the Assumption as:2:first be in force. For any $n \in \mathbb{N}$ and any ${\bf x} \in \mathcal{V}$ the law of ${\bf X}_n^{\bf x}$ is regularly varying. More precisely ${\bf X}_n^{\bf x}\in {\rm Rv}(\alpha, \Theta_{n}^{\bf x})$, where and The processes $\{{\bf X}_n^{\bf x}\}_{n \in \mathbb{N}}$, $\{{\bf Z}_n^{{\bf x}}\}_{n \in \mathbb{N}}$ and $\{\Theta ({\bf y})\}_{{\bf y} \in \mathcal{V}}$

Theorems & Definitions (27)

  • Definition 2.1
  • Proposition 2.3
  • Proposition 2.5
  • Theorem 2.7
  • Theorem 2.10
  • Theorem 2.11
  • Corollary 3.4
  • Corollary 3.5
  • Lemma 4.1
  • proof
  • ...and 17 more