Let X = (X1, X2, …) be a sequence of random variables with values in a standard space (S, B). Suppose(FORMULA PRESENT)where θ > 0 is a constant, ν a probability measure on B, and K a random probability measure on B. Then, X is exchangeable whenever K is a regular conditional distribution for ν given any sub-σ-field of B. Under this assumption, X enjoys all the main properties of classical Dirichlet sequences, including Sethuraman’s representa-tion, conjugacy property, and convergence in total variation of predictive distributions. If μ is the weak limit of the empirical measures, conditions for μ to be a.s. discrete, or a.s. non-atomic, or μ ≪ ν a.s., are provided. Two CLT’s are proved as well. The first deals with stable convergence while the second concerns total variation distance.
CITATION STYLE
Berti, P., Dreassi, E., Leisen, F., Pratelli, L., & Rigo, P. (2023). Kernel based Dirichlet sequences. Bernoulli, 29(2), 1321–1342. https://doi.org/10.3150/22-BEJ1500
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