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 $(z_t)_{z\in\N}$ with given equal one-dimensional marginal distribution, one can always construct a function $f$ and a Gaussian process $(X_t)_{t\in\N}$ such that $\left(f(X_t)\right)_{t\in\N}$ has the same marginal distributions and, asymptotically, the same autocovariance function as $(z_t)_{t\in\N}$. 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.
Highlights
Time series models are of great significance in numerous areas of applications, e.g., finance, climatology, and signal processing, to name just a few
We show that if the one-dimensional marginal distribution is symmetric, the corresponding Hermite ranks for variance and autocovariance estimators are equal to 2
We argued why it is advantageous to model weakly stationary time series with equal one-dimensional marginals by using Gaussian subordinated processes, especially in the case of long memory
Summary
Time series models are of great significance in numerous areas of applications, e.g., finance, climatology, and signal processing, to name just a few. Motivated by real-life applications, the non-central limit theorems have been studied mostly in the case of long-memory processes In this case one has to use stronger normalization, and the limiting distribution is Gaussian only if the so-called Hermite rank of the function f is 1. We obtain limiting normal distributions for the traditional mean and autocovariance estimators for any time series within our model that has absolutely summable autocovariance function This corresponds to the case with short memory. In the short memory case our assumption of summable covariance function is rather intuitive, as well as verified, compared to, e.g., complicated assumptions on the coefficients φj of linear processes These results highlight the applicability of Gaussian subordinated processes in modeling weakly stationary time series.
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