Regenerative Bootstrap for Markov chains in high dimension

Large-dimensional Markov chains appear in many models and many applications. In this paper, we introduce the Multidimensional Approximative Regenerative Block Bootstrap (MARBB), a bootstrap algorithm designed for high-dimensional Markov chains. We focus, in this paper, on a vector autoregressive (VAR(1)) process with a low-rank structure. We first use a reduction algorithm that transforms the original high-dimensional time series into a lower-dimensional Markov chain. Once the chain is reduced, we leverage the regenerative properties of Harris recurrent Markov chains within a general state space, using the Nummelin splitting technique to extend existing results from the one-dimensional settings to the multidimensional case. This approach enables the identification of approximate regeneration times of the lower-dimensional Markov chain, which in turn leads to the splitting of the original high-dimensional Markov chain into regenerative blocks. These blocks are then used to bootstrap relevant statistics based on regenerative blocks. Finally, we give the MARBB consistency results, and we apply our algorithm to simulation data.