Kernel based Dirichlet sequences

Let $X=(X_1,X_2,ldots)$ be a sequence of random variables with values in a standard space $(S,mathcal{B})$. Suppose egin{gather*} X_1sim uquad ext{and}quad Pigl(X_{n+1}incdotmid X_1,ldots,X_nigr)=rac{ heta u(cdot)+sum_{i=1}^nK(X_i)(cdot)}{n+ heta}quadquad ext{a.s.} end{gather*} where $ heta>0$ is a constant, $ u$ 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 $ u$ 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 $mull u$ a.s., are provided. Two CLT's are proved as well. The first deals with stable convergence while the second concerns total variation distance.