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# Gini autocovariance function used for time series with heavy‐tail distributions

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The use of covariance is limited by the need of the finite second moment. This restriction excludes the use of heavy tailed distributions and data. By developing methods that only require finite first moment assumption of the variables we allow for heavy tailed scenarios. Such methods would be valid in the heavy tailed setting, conceptually meaningful in a population model and effective in practical applications. Gini autocovariance function (Gini ACV) is defined under merely finite first moment assumption. Conceptually, it plays a similar role as the usual Pearson autocovariance function, and thus represents a new fundamental tool in time series modeling. The latest applications of Gini ACV and Gini autocorrelation function (Gini ACF) are reviewed, providing a big picture of the latest research in the area. The formulation of Gini ACV and Gini ACF is quickly reviewed and followed by the applications in linear time series modeling, unit root test and reversibility test in time series and nonlinear autoregressive Pareto process. This article is categorized under Statistical Learning and Exploratory Methods of the Data Sciences > Manifold Learning Statistical and Graphical Methods of Data Analysis > Nonparametric Methods Statistical Models > Time Series Models
Sample paths of an AR(1) process of time length 500, for φ = 0.5 and standard normal innovations (upper plot). The lag 25 sample Gini ACF and Pearson ACF (lower two graphs)
[ Normal View | Magnified View ]
Sample paths of an MA(2) process of time length 500, for θ1 = 0.8, θ2 = −0.5 and Pareto(1.5) innovations (upper plot). The lag 25 sample Gini ACF and Pearson ACF (lower two graphs)
[ Normal View | Magnified View ]
Sample paths of an MA(2) process of time length 500, for θ1 = 0.8, θ2 = −0.5 and normal innovations (upper plot). The lag 25 sample Gini ACF and Pearson ACF (lower two graphs)
[ Normal View | Magnified View ]
Sample paths of an AR(2) process of time length 500, for φ1 = 0.8, φ2 = −0.5 and Pareto(1.5) innovations (upper plot). The lag 25 sample Gini ACF and Pearson ACF (lower two graphs)
[ Normal View | Magnified View ]
Sample paths of an AR(2) process of time length 500, for φ1 = 0.8, φ2 = −0.5 and normal innovations (upper plot). The lag 25 sample Gini ACF and Pearson ACF (lower two graphs)
[ Normal View | Magnified View ]
Sample paths of an MA(1) process of time length 500, for θ = 0.5 and Pareto(1.5) innovations (upper plot). The lag 25 sample Gini ACF and Pearson ACF (lower two graphs)
[ Normal View | Magnified View ]
Sample paths of an MA(1) process of time length 500, for θ = 0.5 and standard normal innovations (upper plot). The lag 25 sample Gini ACF and Pearson ACF (lower two graphs)
[ Normal View | Magnified View ]
Sample paths of an AR(1) process of time length 500, for φ = 0.5 and Pareto(1.5) innovations (upper plot). The lag 25 sample Gini ACF and Pearson ACF (lower two graphs)
[ Normal View | Magnified View ]