A Generalization of Stationary AR(1) Schemes

TL;DR

This paper introduces a novel first-order autoregressive model where the process is marginally stationary, involving sums or extremes of i.i.d. observations, and characterizes its stationary solutions in terms of semi-selfdecomposability and stability.

Contribution

It generalizes stationary AR(1) schemes by incorporating sums and extremes, providing a comprehensive characterization of their stationary solutions.

Findings

Stationary solutions are semi-selfdecomposable or extreme-semi-selfdecomposable.

Solutions can also be sum or extreme stable with respect to Harris distribution.

The model broadens the class of stationary autoregressive processes.

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.