Markov chains, AR linear models, and regular variation

TL;DR

This paper explores multivariate regular variation in Markov chains and linear models, showing how the regular variation of stationary distributions relates to innovations and how the structure of linear models influences spectral measures.

Contribution

It establishes conditions under which regular variation of stationary distributions can be derived from innovations and analyzes the impact of chain structure on spectral measures in linear models.

Findings

Regular variation of stationary distribution derived from innovations under monotonicity.

The structure of the chain influences the spectral measure in linear models.

Conditions identified for regular variation in multivariate Markov chains.

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.