Let $p=(p_1,p_2,\ldots)$ be a probability distribution on the positive integers. WLOG, assume $p_1 \ge p_2 \ge \ldots$.
Let $i_1,\ldots,i_T$ be an iid sample from $p$, and for any positive integer $i$, let $n_T(i)$ be the number of times it occurs in the sample, i.e $n_T(i) := \#\{t \mid i_t = i\}$. Sort the $n_T(i)$'s in decreasing order and let $r_T(i)$ be the rank of $i$. For any positive integer $n$, define $$ k_T(n) := \frac{\#\{i \mid r_T(i) \le n\}}{n} = \frac{\sum_i 1[r_T(i) \le n]}{n}. $$
Question. What are good concentration-type lower and upper-bounds for $k_T(n)$ in terms of $T$ and $n$ ?
I'm particularly interested in the case where $p_i \propto i^{-\beta}$ for some constant $\beta \gt 1$.