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Inference on the change point with the jump size near the boundary of the region of detectability in high dimensional time series models.

We develop a projected least squares estimator for the change point parameter in a high dimensional time series model with a potential change point. Importantly we work under the setup where the jump size may be near the boundary of the region of detectability. The proposed methodology yields an optimal rate of convergence despite high dimensionality of the assumed model and a potentially diminishing jump size. The limiting distribution of this estimate is derived, thereby allowing construction of a confidence interval for the location of the change point. A secondary near optimal estimate is proposed which is required for the implementation of the optimal projected least squares estimate. The prestep estimation procedure is designed to also agnostically detect the case where no change point exists, thereby removing the need to pretest for the existence of a change point for the implementation of the inference methodology. Our results are presented under a general positive definite spatial dependence setup, assuming no special structure on this dependence. The proposed methodology is designed to be highly scalable, and applicable to very large data. Theoretical results regarding detection and estimation consistency and the limiting distribution are numerically supported via monte carlo simulations.