  Multivariate time series data that capture the temporal evolution of
interconnected systems are ubiquitous in diverse areas. Understanding the
complex relationships and potential dependencies among co-observed variables is
crucial for the accurate statistical modelling and analysis of such systems.
Here, we introduce kernel-based statistical tests of joint independence in
multivariate time series by extending the $d$-variable Hilbert-Schmidt
independence criterion (dHSIC) to encompass both stationary and non-stationary
processes, thus allowing broader real-world applications. By leveraging
resampling techniques tailored for both single- and multiple-realisation time
series, we show how the method robustly uncovers significant higher-order
dependencies in synthetic examples, including frequency mixing data and logic
gates, as well as real-world climate, neuroscience, and socioeconomic data. Our
method adds to the mathematical toolbox for the analysis of multivariate time
series and can aid in uncovering high-order interactions in data.
