# Conditioned sum of n Poissons versus unconditioned Poissons

Let $\theta >1$ and take independent random variables $Z_k \sim \text{Poisson}(\theta/k)$ for $1 \leq k \leq n$ and let $Z_k^*$ have marginals like the $Z_k$ conditioned on $\sum_1^n k Z_k = n$: $$\mathbf P[Z_k^* = j] = \mathbf P\left [Z_k = j \; \Big| \; \sum_1^n k Z_k =n\right ].$$

This is known as Ewen's measure. Notice that $\mathbf E \sum_1^n k Z_k = \theta n$, hence conditioning this to be $n$ should "squish" the $Z^*_k$, and make them "smaller" than their independent counterparts $Z_k$.

Stochastic dominance $Z^*_k \preceq Z$ does not appear to hold, but it would be sufficient for our purposes to prove a statement like $$\liminf_{n >0} \; \textbf P\left[ \bigcap_{k= 1}^{n} \{Z_k^* \leq Z_k\}\right] > 0.$$

Is there a standard approach for such a bound?

The idea is that the $Z^*_k$ can be coupled to the $Z_k$ so that $$Z_k^* \leq Z_k + 1(J_n = k),$$ for an easy to describe random variable $J_n$. This says that the conditioned $Z_k^*$ are larger than the $Z_k$ in at most one entry.