An urn model with random multiple drawing and random addition

We consider a two-color urn model with multiple drawing and random time-dependent addition matrix. The model is very general with respect to previous literature: the number of sampled balls at each time-step is random, the addition matrix is not balanced and it has general random entries. For the proportion of balls of a given color, we prove almost sure convergence results. In particular, in the case of equal reinforcement averages, we prove fluctuation theorems (through CLTs in the sense of stable convergence and of almost sure conditional convergence, which are stronger than convergence in distribution) and we give asymptotic confidence intervals for the limit proportion, whose distribution is generally unknown.