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I am fairly new to the area of time series and I am trying to understand the notion of long-range dependence in time series. My goal is to characterize the same in the case of multi-variate time series. The Wikipedia page on long-range dependence talks about using some second-order statistics such as auto-covariance, partial sums etc,. for detecting long-term memory. Is there any way these methods can be generalized to multi-variate case, i.e at each time $t$, my time series value $x_t \in \mathbb{R}^d$?

Is there an agreed notion of how to test long-range dependency even for scalar valued time series?

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