Abstract
Under Poincare-type conditions, upper bounds are explored for the Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law. Based on improved concentration inequalities on high-dimensional Euclidean spheres, the results extend and refine previous results to non-symmetric models.
Highlights
Let X = (X1, . . . , Xn) be an isotropic random vector in Rn (n ≥ 2), meaning that EXiXj = δij for all i, j ≤ n, where δij is the Kronecker symbol
We are looking for natural general conditions on Xk which guarantee that the distribution functions Fθ(x) = P{Sθ ≤ x} are well approximated for most of θ ∈ Sn−1 by the standard normal distribution function
Of special interest is the question of possible rates in the Kolmogorov distance ρ(Fθ, Φ) = sup |Fθ(x) − Φ(x)|
Summary
Modulo n-dependent logarithmic factors, the following three assertions are equivalent up to positive constants c and β (perhaps different in different places) for the entire class of isotropic random vectors X in Rn having symmetric log-concave distributions (cf [13]):. In this connection, let us mention a recent paper by Jiang, Lee and Vempala [22], which provides a reformulation of (i)-(ii) as a central limit theorem for random variables of the form X, Y , where Y is an independent copy of X. We denote by c a positive absolute constant which may vary from place to place (if not stated explicitly that c depends on some parameter)
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