Kernel based Dirichlet sequences

Let X=(X1,X2,…) be a sequence of random variables with values in standard space (S,B). Suppose X1∼νandP(Xn+1∈⋅∣X1,…,Xn)=θν(⋅)+ ∑i=1nK(Xi)(⋅) n+θa.s. where θ>0 is constant, ν probability measure on B, and K B. Then, X exchangeable whenever regular conditional distribution for given any sub-σ-field Under this assumption, enjoys all the main properties classical Dirichlet sequences, including Sethuraman’s representation, conjugacy property, convergence total variation predictive distributions. If μ weak limit empirical measures, conditions to a.s. discrete, or non-atomic, μ≪ν a.s., are provided. Two CLT’s proved as well. The first deals stable while second concerns distance.