Order 1 autoregressive process of finite length

TL;DR

This paper compares finite and infinite length order 1 autoregressive processes, highlighting how finite length affects properties and transient behaviors, especially with significant serial correlation, and proposes a method to avoid transients.

Contribution

It provides a theoretical comparison between finite and infinite order 1 autoregressive processes and introduces a way to prevent transient effects in finite processes.

Findings

Finite length processes exhibit transient effects with significant serial correlation.

Choosing the initial term as a Gaussian variable with specific standard deviation avoids transients.

Properties of finite and infinite processes differ mainly when serial correlation is strong.

Abstract

The stochastic processes of finite length defined by recurrence relations request additional relations specifying the first terms of the process analogously to the initial conditions for the differential equations. As a general rule, in time series theory one analyzes only stochastic processes of infinite length which need no such initial conditions and their properties are less difficult to be determined. In this paper we compare the properties of the order 1 autoregressive processes of finite and infinite length and we prove that the time series length has an important influence mainly if the serial correlation is significant. These different properties can manifest themselves as transient effects produced when a time series is numerically generated. We show that for an order 1 autoregressive process the transient behavior can be avoided if the first term is a Gaussian random variable with standard deviation equal to that of the theoretical infinite process and not to that of the white noise innovation.