Uniform versus Zipf distribution in a mixing collection process

We consider the following variant of the classic collector's problem: The family of coupon probabilities is the mixing of two subfamilies one of which is the \textit{uniform} family, while the other belongs to the well known \textit{Zipf family}. We obtain asymptotics for the expectation, the second rising moment, and the variance of the random variable $T_N$, namely the number of trials needed for all the $N$ types of coupons to be collected (at least once, with replacement) as $N \rightarrow \infty$. It is interesting that the effect of the uniform subcollection on the asymptotics of the expectation of $T_N$ (at least up to the sixth term) appears only in the leading factor of the expectation of $T_N$. The limiting distribution of $T_N$ is derived as well. These results answer a question placed in a recent work of ours [\textit{Electron. J. Probab.} \textbf{18} (2012) 1--15].