On modeling weakly stationary processes

2017/07/29 by Lauri Viitasaari, Pauliina Ilmonen, Viitasaari, Lauri +1
Economics, Econometrics and Finance · Mathematics · #60F05 #60G10 #62M10 #Complex Systems and Time Series Analysis #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Statistical Methods and Inference

paper · pdf · doi:10.48550/arxiv.1707.09490

openalex publication_date 2017/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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∈ℕ with given equal one-dimensional marginal distribution, one can always construct a function f and a Gaussian process (X_t)_t∈ℕ such that (f(X_t))_t∈ℕ has the same marginal distributions and, asymptotically, the same autocovariance function as (z_t)_t∈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.

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