Negative dependence in sampling

The strong Rayleigh property is a new and robust negative dependence property that implies negative association; in fact it implies conditional negative association closed under external fields (CNA+). Suppose thatX1, . . . , XN and Y1, . . . , YN are two families of 0-1 random variables that satisfy the strong Rayleigh property and let Zi := Xi + Yi. We show that {Zi} conditioned on ∑ iXiYi = 0 is also strongly Rayleigh; this turns out to be an easy consequence of the results on preservation of stability of polynomials of Borcea & Brändén (2009). This entails that a number of important πps sampling algorithms, including Sampford sampling and Pareto sampling, are CNA+. As a consequence, statistics based on such samples automatically satisfy a version of the Central Limit Theorem for triangular arrays. AMS Subject classification : 60J10