[Paper Review] Multivariate regular variation of heavy-tailed Markov chains

Johan Segers|ArXiv.org|Jan 15, 2007

Stochastic processes and statistical mechanics18 references19 citations

TL;DR

This paper establishes the multivariate regular variation of heavy-tailed Markov chains by introducing a back-and-forth tail chain framework that captures the joint asymptotic behavior of forward and backward extremes. It shows that under mild regular variation and scaling conditions on the transition mechanism, the finite-dimensional distributions of the process are multivariate regularly varying, even when tail switching between positive and negative extremes occurs.

ABSTRACT

The upper extremes of a Markov chain with regulary varying stationary marginal distribution are known to exhibit under general conditions a multiplicative random walk structure called the tail chain. More generally, if the Markov chain is allowed to switch from positive to negative extremes or vice versa, the distribution of the tail chain increment may depend on the sign of the tail chain on the previous step. But even then, the forward and backward tail chain mutually determine each other through a kind of adjoint relation. As a consequence, the finite-dimensional distributions of the Markov chain are multivariate regularly varying in a way determined by the back-and-forth tail chain. An application of the theory yields the asymptotic distribution of the past and the future of the solution to a stochastic difference equation conditionally on the present value being large in absolute value.

Motivation & Objective

• To characterize the weak limits of finite-dimensional distributions of a heavy-tailed Markov chain conditionally on the present value being large in absolute value.
• To extend the classical tail chain representation to cases where extreme values can switch between upper and lower tails, which invalidates standard i.i.d. multiplicative random walk models.
• To establish a formal adjoint relationship between forward and backward tail chains in stationary Markov chains, enabling a joint asymptotic description.
• To demonstrate that under general conditions, the full process is multivariate regularly varying, with dependence structure fully determined by the back-and-forth tail chain.
• To apply the theory to stochastic difference equations, particularly stationary solutions of stochastic recurrence equations with heavy-tailed innovations.

Proposed method

• Define the forward tail chain as the weak limit of the rescaled process $(X_0, X_1, \ldots, X_t)$ given $|X_0| > x$ as $x \to \infty$, under regular variation of the marginal distribution.
• Introduce a generalized tail-switching mechanism where the increment distribution depends on the sign of the previous state, leading to a non-i.i.d. tail chain.
• Establish an adjoint relation between forward and backward tail chains via a bivariate transformation, allowing mutual determination of the two chains.
• Use the back-and-forth tail chain to characterize the joint multivariate regular variation of the full process, even when the forward chain is not a simple random walk.
• Derive conditions on the recursive function $\Psi$ in the stochastic recurrence $X_t = \Psi(X_{t-1}, \varepsilon_t)$ ensuring the existence of the forward and backward tail chains.
• Apply the framework to examples such as stochastic difference equations and max-autoregressive processes with latent states, showing how path dependence and regime switching affect tail behavior.

Experimental results

Research questions

• RQ1 How can the asymptotic behavior of a heavy-tailed Markov chain be characterized when extreme values can switch between positive and negative tails?
• RQ2 What conditions on the recursive mechanism $\Psi$ ensure the existence of a well-defined forward tail chain, even when the increment distribution depends on the sign of the previous state?
• RQ3 How are the forward and backward tail chains related in a stationary Markov chain, and can this relationship be formalized as an adjoint structure?
• RQ4 Under what conditions is the finite-dimensional distribution of the process multivariate regularly varying, and how is this determined by the back-and-forth tail chain?
• RQ5 Can the theory be applied to model the joint tail behavior of solutions to stochastic difference equations, particularly those with heavy-tailed innovations and time-varying dynamics?

Key findings

• The forward tail chain of a heavy-tailed Markov chain converges weakly to a multiplicative process with increments depending on the sign of the previous state, generalizing the classical i.i.d. random walk structure.
• In the stationary case, the forward and backward tail chains are linked via an adjoint relation, forming a back-and-forth tail chain that fully characterizes the multivariate regular variation of the process.
• The finite-dimensional distributions of the Markov chain are multivariate regularly varying, with the dependence structure determined by the joint forward and backward tail chains.
• For a stationary solution of a stochastic difference equation with regularly varying marginal distribution, the conditional distribution of past and future values given a large present value converges to a distribution determined by the back-and-forth tail chain.
• Counterexamples show that the theory breaks down when the transition mechanism does not satisfy the asymptotic scaling condition (2.2), particularly when the tail behavior of the innovation depends on the state in a non-asymptotically stable way.
• In cases with latent Markov switching (e.g., max-AR processes), the tail chain increments are not independent, and the forward tail chain is not a simple i.i.d. multiplicative random walk, even if the marginal law is regularly varying.