AUTHOR=Viitasaari Lauri , Ilmonen Pauliina 

TITLE=On Modeling a Class of Weakly Stationary Processes

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=5

YEAR=2020

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2019.00068

DOI=10.3389/fams.2019.00068

ISSN=2297-4687

ABSTRACT=<p>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 (<italic>z</italic><sub><italic>t</italic></sub>)<sub><italic>z</italic>∈ℕ</sub> with given equal one-dimensional marginal distribution, one can always construct a function <italic>f</italic> and a Gaussian process (<italic>X</italic><sub><italic>t</italic></sub>)<sub><italic>t</italic>∈ℕ</sub> such that (<italic>f</italic>(<italic>X</italic><sub><italic>t</italic></sub>))<sub><italic>t</italic>∈ℕ</sub> has the same marginal distributions and, asymptotically, the same autocovariance function as (<italic>z</italic><sub><italic>t</italic></sub>)<sub><italic>t</italic>∈ℕ</sub>. 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.</p>