On modeling weakly stationary processes

TL;DR

This paper demonstrates that weakly stationary processes can be effectively modeled using Gaussian subordinated processes, providing a flexible alternative to linear models with weaker assumptions.

Contribution

It introduces a method to model any weakly stationary time series via Gaussian subordinated processes, enabling asymptotic analysis of estimators.

Findings

Gaussian subordinated processes can replicate marginal distributions.

Asymptotic distributions for estimators are derived.

Model offers greater flexibility than standard linear models.

Abstract

In this article, we show that a general class of weakly stationary time series can be modeled applying Gaussian subordinated processes. We show that, for any given weakly stationary time series $(z_t)_{z\in\mathbb{N}}$ with given equal one-dimensional marginal distribution, one can always construct a function $f$ and a Gaussian process $(X_t)_{t\in\mathbb{N}}$ such that $\left(f(X_t)\right)_{t\in\mathbb{N}}$ has the same marginal distributions and, asymptotically, the same autocovariance function as $(z_t)_{t\in\mathbb{N}}$. Consequently, we obtain asymptotic distributions for the mean and autocovariance estimators by using the rich theory on limit theorems for Gaussian subordinated processes. This highlights the role of Gaussian subordinated processes in modeling general weakly stationary time series. We compare our approach to standard linear models, and show that our model is more flexible and requires weaker assumptions.