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The increased demand for online prediction and the growing availability of large data sets drives the need for computationally efficient models. While exact Gaussian process regression shows various favorable theoretical properties (uncertainty estimate, unlimited expressive power), the poor scaling with respect to the training set size prohibits its application in big data regimes in real-time. Therefore, this paper proposes dividing local Gaussian processes, which are a novel, computationally efficient modeling approach based on Gaussian process regression. Due to an iterative, data-driven division of the input space, they achieve a sublinear computational complexity in the total number of training points in practice, while providing excellent predictive distributions. A numerical evaluation on real-world data sets shows their advantages over other state-of-the-art methods in terms of accuracy as well as prediction and update speed.