In the generalized Coupon Collector's Problem, there are $N$ types of coupon, and for any $i \in [N] := \{1,2,\ldots,N\}$, $p_i \ge 0$ is the probability of obtaining a type-i coupon on any trial. Thus, $\sum_{i=1}^N p_i = 1$. Let $T_N$ be the number of (independent) trials required to collect all $N$ types of coupons. Note that $T_N$ is a random integer. Under general regularity conditions on the $p_i$'s (e.g uniformity $p_i = 1/N$ for all $i$; etc.), there exists large-deviation inequalities (LDIs) of the form
$$
\mathbb P(T_N \le b_N + zk_N) = e^{-e^{-z}} + o_N(1),
$$
for some appropriate $b_N$ and $k_N$ with $k_N/b_N = o(1)$. See the paper "The Coupon Collector's Problem Revisited: Generalizing the Double Dixie Cup Problem of Newman and Shepp" https://arxiv.org/abs/1412.3626 for details.

For example, in the uniform case where $p_i = 1/N$ for all $i$, it is well-known that $b_N = N\log N$ and $k_N = N$.

I'm interested in another quantity. Define $N_T$ to be the number of coupon types collected after $T$ trials. It is clear that $N_T = N$ iff $T_N \le T$. I'm interested in lower obounds (and perhaps bounds too) for $\mathbb P(N_T \ge n)$ for any $n=1,2,\ldots,N$.

**Question.** Are the large deviation inequalities for $N_T$ (perhaps for appropriately scaled $T$) ?

**N.B.:** I'm particularly interested in the case where the $p_i$'s a rapidly decreasing, e.g $p_i \propto i^{-b}$ for some constant $b \gt 1$.

Observations
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- $\mathbb E\, N_T = N - \sum_{i=1}^N (1-p_i)^T \ge N-\sum_{i=1}^N e^{-p_i T}$. Indeed, let $n_T(i)$ be the number of times a type-i coupon is observed in $T$ trials. Then, $N_T = \sum_{i=1}^N 1[n_T(i) \ge 1]$. By linearity of expectation and the fact that
$$
\mathbb E 1[n_T(i) \ge 1] = \mathbb P(n_T(i) \ge 1) = 1 - \mathbb P(n_T(i) = 0) = 1 - (1-p_i)^T \ge 1-e^{-p_iT},
$$
 the claim follows.