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Conditional negative association for competing urns

Competing urns refers to the random experiment where m balls are dropped, randomly and independently, into urns 1,...,n. Formally, we have a random map $σ$ from {1,...,m} to {1,...,n} with the $σ(i)$'s i.i.d. With $x_j$ the indicator of the event that at least $t_j$ balls land in urn j (for some threshold $t_j$), we prove conditional negative association for the random variables $x_1,...,x_n$. We mostly deal with the more general situation in which the $σ(i)$'s need not be identically distributed, proving results which imply conditional negative association in the i.i.d. case. Some of the results--particularly Lemma 8 on graph orientations--are thought to be of independent interest. We also give a counterexample to a negative correlation conjecture of D. Welsh, a strong version of a (still open) conjecture of G. Farr.