Quantitative asymptotics of graphical projection pursuit

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There is a result of Diaconis and Freedman which says that, in a limiting sense, for large collections of high-dimensional data most one-dimensional projections of the data are approximately Gaussian. This paper gives quantitative versions of that result. For a set of deterministic vectors {xᵢ}ᵢ₌₁ⁿ in ᵈ with n and d fixed, let θ∈ᵈ⁻¹ be a random point of the sphere and let μₙθ denote the random measure which puts mass 1/n at each of the points x₁θ,...,xₙθ. For a fixed bounded Lipschitz test function f, Z a standard Gaussian random variable and σ² a suitable constant, an explicit bound is derived for the quantity ¶[|∫ f dμₙθ- f(σZ)|>ε]. A bound is also given for ¶[dBL(μₙθ, N(0,σ²))>ε], where dBL denotes the bounded-Lipschitz distance, which yields a lower bound on the waiting time to finding a non-Gaussian projection of the {xᵢ} if directions are tried independently and uniformly on ᵈ⁻¹.