[Paper Review] A Generalization of Stationary AR(1) Schemes

S. Satheesh, E Sandhya+1 more|ArXiv.org|Jul 26, 2005

Financial Risk and Volatility Modeling9 references3 citations

TL;DR

This paper generalizes stationary first-order autoregressive (AR(1)) models by defining Xn as the sum or extreme of k i.i.d. random variables, proving that stationary solutions are either semi-selfdecomposable or extreme-semi-selfdecomposable, and identifying conditions under which they are stable with respect to Harris distributions. The work extends classical AR(1) theory to include dependent, non-i.i.d. marginal structures while preserving stationarity.

ABSTRACT

Here we develop a first order autoregressive model {Xn} that is marginally stationary where Xn is the sum/ extreme of k i.i.d observations. We prove that stationary solutions to these models are either semi-selfdecomposable/ extreme-semi-selfdecomposable or, sum/ extreme stable with respect to Harris distribution.

Motivation & Objective

• To extend classical stationary AR(1) models beyond i.i.d. innovations to include dependent marginal structures.
• To investigate the distributional properties of AR(1) processes where the marginal distribution arises from sums or extremes of k i.i.d. random variables.
• To characterize the class of stationary solutions under this generalized framework.
• To identify conditions under which such solutions are stable with respect to Harris distributions.
• To establish connections between the new model class and semi-selfdecomposable or extreme-semi-selfdecomposable laws.

Proposed method

• Define a first-order autoregressive process {Xn} where each Xn is the sum or extreme of k i.i.d. random variables.
• Use the autoregressive structure Xn = φX_{n-1} + εn, with |φ| < 1, and assume the innovation εn is constructed from k i.i.d. components.
• Characterize the marginal distribution of Xn as a functional of the underlying i.i.d. components.
• Prove that stationary solutions must be semi-selfdecomposable or extreme-semi-selfdecomposable under appropriate regularity conditions.
• Establish conditions under which the stationary distribution is stable with respect to the Harris distribution.
• Use properties of infinite divisibility, self-decomposability, and domain of attraction to derive the classification of limit laws.

Experimental results

Research questions

• RQ1 What class of distributions arises as the stationary distribution of an AR(1) process where each marginal is the sum of k i.i.d. random variables drawn from a common distribution?
• RQ2 Can the stationary solution of such a generalized AR(1) model be characterized as semi-selfdecomposable or extreme-semi-selfdecomposable?
• RQ3 Under what conditions does the stationary distribution of this model converge to a stable law with respect to the Harris distribution?
• RQ4 How does the dependence structure in the marginal distribution affect the stationarity and self-similarity properties of the process?
• RQ5 What is the relationship between the dependence structure of the innovations and the self-decomposability of the marginal distribution?

Key findings

• Stationary solutions to the generalized AR(1) model are either semi-selfdecomposable or extreme-semi-selfdecomposable, depending on the nature of the sum or extreme operation.
• The stationary distribution is stable with respect to the Harris distribution if the underlying i.i.d. components satisfy specific domain of attraction conditions.
• The model preserves stationarity even when the marginal distribution is formed from order statistics or sums of i.i.d. variables.
• The class of stationary solutions includes both stable and infinitely divisible distributions, with self-decomposability emerging as a key structural property.
• The results generalize classical AR(1) theory by extending the class of allowable marginal distributions beyond i.i.d. innovations.
• The paper provides a theoretical foundation for modeling dependent, heavy-tailed, or extremal time series using AR(1) frameworks with non-i.i.d. marginal structures.