  We develop two models for the temporal evolution of extreme events of
multivariate $k$th order Markov processes. The foundation of our methodology
lies in the conditional extremes model of Heffernan & Tawn (2004), and it
naturally extends the work of Winter & Tawn (2016,2017) and Tendijck et al.
(2019) to include multivariate random variables. We use cross-validation-type
techniques to develop a model order selection procedure, and we test our models
on two-dimensional meteorological-oceanographic data with directional
covariates for a location in the northern North Sea. We conclude that the
newly-developed models perform better than the widely used historical matching
methodology for these data.
