Negative dependence in sampling. (English) Zbl 1319.62018
Summary: 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 that \(X_1, \ldots, X_N\) and \(Y_1, \ldots, Y_N\) are two families of 0-1 random variables that satisfy the strong Rayleigh property and let \(Z_i:=X_i + Y_i\). We show that \(\{Z_i\}\) conditioned on \(\sum_i X_iY_i=0\) is also strongly Rayleigh; this turns out to be an easy consequence of the results on preservation of stability of polynomials of J. Borcea and P. Brändén [Invent. Math. 177, No. 3, 541–569 (2009; Zbl 1175.47032)]. This entails that a number of important \(\pi \)ps 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.
Citations:
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