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We develop two models for the temporal evolution of extreme events ofmultivariate $k$th order Markov processes. The foundation of our methodologylies in the conditional extremes model of Heffernan & Tawn (2004), and itnaturally extends the work of Winter & Tawn (2016,2017) and Tendijck et al.(2019) to include multivariate random variables. We use cross-validation-typetechniques to develop a model order selection procedure, and we test our modelson two-dimensional meteorological-oceanographic data with directionalcovariates for a location in the northern North Sea. We conclude that thenewly-developed models perform better than the widely used historical matchingmethodology for these data.

Temporal evolution of the extreme excursions of multivariate $k$th order Markov processes with application to oceanographic data