Correct ordering in the Zipf-Poisson ensemble

We consider a Zipf–Poisson ensemble in which $X_i\sim \text{poi}(N{i}^{-\alpha })$ for $\alpha >1$ and $N>0$ and integers $i\ge 1$. As $N\to \mathrm{\infty }$ the first $n^{\mathrm{\prime }}(N)$ random variables have their proper order $X_1>X_2>\mathrm{.}\mathrm{.}\mathrm{.}>X_{n^{\mathrm{\prime }}}$ relative to each other, with probability tending to 1 for $n^{\mathrm{\prime }}$ up to $(AN\mathrm{/}\log (N){)}^{1\mathrm{/}(\alpha +2)}$ for an explicit constant $A(\alpha )\ge 3\mathrm{/}4$. The rate $N^{1\mathrm{/}(\alpha +2)}$ cannot be achieved. The ordering of the first $n^{\mathrm{\prime }}(N)$ entities does not preclude $X_m>X_{n^{\mathrm{\prime }}}$ for some interloping $m>n^{\mathrm{\prime }}$. The first $n\mathrm{"}$ random variables are correctly ordered exclusive of any interlopers, with probability tending to 1 if $n\mathrm{"}\le (BN\mathrm{/}\log (N){)}^{1\mathrm{/}(\alpha +2)}$ for $B<A$. For a Zipf–Poisson model of the British National Corpus, which has a total word count of $\mathrm{100,000,000}$, our result estimates that the 72 words with the highest counts are properly ordered.