Quantitative asymptotics of graphical projection pursuit

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 $\{x_i\}_{i=1}^n$ in $\R^d$ with $n$ and $d$ fixed, let $θ\in\s^{d-1}$ be a random point of the sphere and let $μ_n^θ$ denote the random measure which puts mass $\frac{1}{n}$ at each of the points $\inprod{x_1}θ,...,\inprod{x_n}θ$. For a fixed bounded Lipschitz test function $f$, $Z$ a standard Gaussian random variable and $σ^2$ a suitable constant, an explicit bound is derived for the quantity $\ds¶[|\int f dμ_n^θ-\E f(σZ)|>ε]$. A bound is also given for $\ds¶[d_{BL}(μ_n^θ, N(0,σ^2))>ε]$, where $d_{BL}$ denotes the bounded-Lipschitz distance, which yields a lower bound on the waiting time to finding a non-Gaussian projection of the $\{x_i\}$ if directions are tried independently and uniformly on $\s^{d-1}$.

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