Berry-esseen Bounds for Projections of Coordinate Symmetric Random Vectors

For a coordinate symmetric random vector $(Y_1,\ldots,Y_n)={\bf Y} \in \mathbb{R}^n$, that is, one satisfying $(Y_1,\ldots,Y_n)=_d(e_1Y_1,\ldots,e_nY_n)$ for all $(e_1,\ldots,e_n) \in \{-1,1\}^n$, for which $P(Y_i=0)=0$ for all $i=1,2,\ldots,n$, the following Berry Esseen bound to the cumulative standard normal $\Phi$ for the standardized projection $W_\theta=Y_\theta/v_\theta$ of ${\bf Y}$ holds: $$ \sup_{x \in \mathbb{R}}|P(W_\theta \leq x) - \Phi(x)| \leq 2 \sum_{i=1}^n |\theta_i|^3 E| X_i|^3 + 8.4 E(V_\theta^2-1)^2, $$ where $Y_\theta=\theta \cdot {\bf Y}$ is the projection of ${\bf Y}$ in direction $\theta \in \mathbb{R}^n$ with $||\theta||=1$, $v_\theta=\sqrt{\mbox{Var}(Y_\theta)},X_i=|Y_i|/v_\theta$ and $V_\theta=\sum_{i=1}^n \theta_i^2 X_i^2$. As such coordinate symmetry arises in the study of projections of vectors chosen uniformly from the surface of convex bodies which have symmetries with respect to the coordinate planes, the main result is applied to a class of coordinate symmetric vectors which includes cone measure ${\cal C}_p^n$ on the $\ell_p^n$ sphere as a special case, resulting in a bound of order $\sum_{i=1}^n |\theta_i|^3$.

where As such coordinate symmetry arises in the study of projections of vectors chosen uniformly from the surface of convex bodies which have symmetries with respect to the coordinate planes, the main result is applied to a class of coordinate symmetric vectors which includes cone measure n p on the ℓ n p sphere as a special case, resulting in a bound of order

Introduction and main result
Properties of the distributions of vectors uniformly distributed over the surface, or interior, of compact, convex bodies, such as the unit sphere in n , have been well studied.When the convex body has symmetry with respect to all n coordinate planes, a vector Y chosen uniformly from its surface satisfies and is said to be coordinate symmetric.Projections of Y along θ ∈ n with ||θ || = 1 have generated special interest, and in many cases normal approximations, and error bounds, can be derived for W θ , the projection Y θ standardized to have mean zero and variance 1.In this note we show that when a random vector is coordinate symmetric, even though its components may be dependent, results for independent random variables may be applied to derive error bounds to the normal for its standardized projection.Bounds in the Kolmogorov and total variation metric for projections of vectors with symmetries are given also in [8], but the bounds are not optimal; the bounds provided here, in particular those in Theorem 2.1 for the normalized projections of the generalization n p,F of cone measure, are of order In related work, many authors study the measure of the set of directions on the unit sphere along which projections are approximately normally distributed, but in most cases bounds are not provided; see in particular [12], [1] and [2].One exception is [6] where the surprising order n i=1 |θ i | 4 is obtained under the additional assumption that a joint density function of Y exists, and is log-concave.When the components Y 1 , . . ., Y n of a coordinate symmetric vector Y have finite variances v 2 1 , . . ., v 2 n , respectively, it follows easily from and hence, that Standardizing to variance 1, write , the common variance of the components, for all θ with ||θ || = 1.One conclusion of Theorem 1.1 gives a Kolmogorov distance bound between the standardized projection W θ and the normal in terms of expectations of functions of We apply Theorem 1.1 to standardized projections of a family of coordinate symmetric random vectors, generalizing cone measure n p on the sphere S(ℓ n p ), defined as follows.With p > 0, let With µ n Lebesgue measure on n , the cone measure of A ⊂ S(ℓ n p ) is given by 3) The cases p = 1 and p = 2 are of special interest, corresponding to the uniform distribution over the unit simplex and unit sphere, respectively.In particular, the authors of [4] compute bounds for the total variation distance between the normal and the components of Y in the case p = 2.
In [5] an L 1 bound between the standardized variable W θ in (1.2) and the normal is obtained when Y has the cone measure distribution.Here an application of Theorem 1.1 yields Theorem 2.1, which gives Kolmogorov distance bounds of the order n i=1 |θ i | 3 for a class of distributions n p,F which include cone measure as a special case.We note that if θ ∈ n satisfies ||θ || = 1, so Hölder's inequality with 1/s + 1/t = 1 yields (1.4) In particular, with s = 3/2, t = 3 we have n −1/2 ≤ n i=1 |θ i | 3 , and therefore, for any sequence of norm one vectors θ in n for n = 1, 2, . . .we have ) for all β > 1/2.We note that equality is achieved in (1.4) when θ = n −1/2 (1, 1, . . ., 1), the case recovering the standardized sum of the coordinates of Y.We have the following simple yet crucial result, shown in Section 3. ) In particular, We remark that in related work, Theorem 4 in [3] gives an exponential non-uniform Berry-Esseen bound for the Studentized sums A simplification of the bounds in Theorem 1.1 result when Y has the 'square negative correlation property,' see [9], that is, when as then and hence the first term on the right hand side of (1.7) can be replaced by 8.4 [9] shows that cone measure n p satisfies a correlation condition much stronger than (1.8); see also [1] regarding negative correlation in the interior of B(ℓ n p ).

Application
One application of Theorem 1.1 concerns the following generalization of cone measure n p .Let n ≥ 2 and G 1 , . . ., G n be i.i.d.nontrivial, positive random variables with distribution function F , and set In addition, let ǫ 1 , . . ., ǫ n be i.i.d.random variables, independent of G 1 , . . ., G n , taking values uniformly in {−1, 1}.Let n p,F be the distribution of the vector By results in [10], for instance, cone measure n p as given in (1.3) is the special case when F is the Gamma distribution Γ(1/p, 1).THEOREM 2.1.Let Y have distribution n p,F given by (2.1) with p > 0 and F for which EG 2+4/p 1 < ∞ when G 1 is distributed according to F .Then there exists a constant c p,F depending on p and F such that for all θ ∈ n for which ||θ || = 1 we have where As the Gamma distribution Γ(1/p, 1) has moments of all orders, the conclusion of Theorem 2.1 holds, in particular, for cone measure n p .
Then, using the coordinate symmetry property to obtain the fourth equality, we have Before proving Theorem 1.1, we invoke the following well-known Berry-Esseen bound for independent random variables (see [11]) In where (3.2) As to R 2 , letting Z ∼ N (0, 1) be independent of V θ we have By monotonicity, it is easy to see that Collecting the bounds above yields as desired.Lastly, (1.7) follows from (1.6) and the fact that Proof of Theorem 2.1.Let Y be distributed as n p,F .With r = 1/p for convenience, first we claim that where the implicit constant in the order here, and below, may depend on p and F .For r ≥ 1/2 Jensen's inequality yields For 0 < r < 1/2, we apply the following exponential inequality for non-negative independent random variables (see, for example, Theorem 2.19 in [7]): obtaining the final inequality by applying (3.7) with n = 1 to the first factor and Chebyshev's inequality to the second.This proves (3.5).
As n p,F is coordinate symmetric with exchangeable coordinates, we apply Theorem 1.1 with v θ = v n as in (3.6), and claim that it suffices to show In particular, regarding the second term in (1.7), we have by (3.5) and (3.8) which dominates (3.9), the order of the first term in (1.7), thus yielding the theorem.Letting µ = EG 1 , the main idea is to use (i) that G 1,n /n is close to µ with probability one by the law of large numbers; and (ii) the Taylor expansions where As to (3.8), again applying (3.14), we have Now, to prove (3.9), write For the next term in (3.19) observe that [by (3.11) and (3.As to R 7 , here using the assumption that EG 2+4r 1 < ∞, using the Cauchy-Schwarz inequality for the second step, we have