L bounds for a central limit theorem with involutions
نویسنده
چکیده
Let E = ((eij))n×n be a fixed array of real numbers such that eij = eji, eii = 0 for 1 ≤ i, j ≤ n. Let the permutation group be denoted by Sn and the collection of involutions with no fixed points by Πn, that is, Πn = {π ∈ Sn : π = id, π(i) 6= i∀i} with id denoting the identity permutation. For π uniformly chosen from Πn, let YE = ∑n i=1 eiπ(i) and W = (YE − μE)/σE where μE = E(YE) and σ 2 E = Var(YE). Denoting by FW and Φ the distribution functions of W and a N (0, 1) variate respectively, we bound ||FW −Φ||p for 1 ≤ p ≤ ∞ using Stein’s method and the zero bias transformation. Optimal Berry-Esseen or L∞ bounds for the classical problem where π is chosen uniformly from Sn were obtained by Bolthausen using Stein’s method. Although in our case π ∈ Πn uniformly, the L bounds we obtain are of similar form as Bolthausen’s bound which holds for p = ∞. The difficulty in extending Bolthausen’s method from Sn to Πn arising due to the involution restriction is tackled by the use of zero bias transformations.
منابع مشابه
bounds for a combinatorial central limit theorem with involutions
Let E = ((eij))n×n be a fixed array of real numbers such that eij = eji, eii = 0 for 1 ≤ i, j ≤ n. Let the symmetric group be denoted by Sn and the collection of involutions with no fixed points by Πn, that is, Πn = {π ∈ Sn : π 2 = id, π(i) 6= i∀i}. For π uniformly chosen from Πn, let YE = Pn i=1 eiπ(i) and W = (YE − μE)/σE where μE = E(YE) and σ 2 E = Var(YE). Denoting by FW and Φ the distribu...
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