2021/05/31 by Berti, Patrizia, Dreassi, Emanuela, Leisen, Fabrizio +2
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2106.00114
Let X=(X1,X2,…) be a sequence of random variables with values in a standard space (S,B). Suppose X1∼ν\quadand P(Xn+1∈⋅| X1,…,Xn)=\fracθν(⋅)+∑i=1nK(Xi)(⋅)n+θ \quada.s. 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 representation, 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.