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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
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Kolmogorov doesn't show existence of Dirichlet process for arbitrary measurable spaces. Why?
Constructing the Dirichlet process using Kolmogorov raises several problems, and your question shows that you are already on to some of them. In short, an arbitrary measurable space is not sufficient, …