Remarks on multivariate Gaussian Process
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Gaussian process occupies one of the leading places in modern Statistics and Probability due to its importance and wealth of results. The common use of GP is to connect with problems related to estimation, detection, and even many statistical or machine learning models. With the fast development of Gaussian process applications, it is necessary to consolidate the fundamentals of vector-valued stochastic process, in particular, multivariate Gaussian process, which is the essential theory for many application problems with multiple correlated responses. In this paper, we proposed a proper definition of multivariate Gaussian process based on Gaussian measure on vector-valued function space and provided its proof of existence. In addition, several fundamental properties of multivariate Gaussian process such as strict stationarity and independence are introduced. We further derived multivariate pre-Brownian motion as a special case of multivariate Gaussian process and presented a brief introduction of multivariate Gaussian process regression as a useful statistical learning method for multi-output prediction problems.
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