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Berti, Patrizia; Dreassi, Emanuela; Leisen, Fabrizio; Pratelli, Luca; Rigo, Pietro

Kernel based Dirichlet sequences. (English) Zbl 1527.60003

Bernoulli 29, No. 2, 1321-1342 (2023).
Summary: Let \(X = (X_1, X_2, \dots)\) be a sequence of random variables with values in a standard space \((S, \mathcal{B})\). Suppose \[ X_1 \sim \nu \quad \text{and} \quad P(X_{n+1} \in \cdot \mid X_1, \dots, X_n) = \frac{\theta \nu (\cdot) + \sum_{i=1}^n K(X_i)(\cdot)}{n + \theta} \qquad \text{a.s.} \] where \(\theta > 0\) is a constant, \(\nu\) a probability measure on \(\mathcal{B}\), and \(K\) a random probability measure on \(\mathcal{B}\). Then, \(X\) is exchangeable whenever \(K\) is a regular conditional distribution for \(\nu\) given any sub-\(\sigma\)-field of \(\mathcal{B}\). Under this assumption, \(X\) enjoys all the main properties of classical Dirichlet sequences, including Sethuraman’s representation, conjugacy property, and convergence in total variation of predictive distributions. If \(\mu\) is the weak limit of the empirical measures, conditions for \(\mu\) to be a.s. discrete, or a.s. non-atomic, or \(\mu \ll \nu\) a.s., are provided. Two CLT’s are proved as well. The first deals with stable convergence while the second concerns total variation distance.

Cited in 2 Documents

MSC:

60B10 Convergence of probability measures
60G09 Exchangeability for stochastic processes
60G57 Random measures
62G05 Nonparametric estimation
60F05 Central limit and other weak theorems

Keywords:

Bayesian nonparametrics; central limit theorem; Dirichlet sequence; exchangeability; predictive distribution; random probability measure; regular conditional distribution
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References:

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