Stein’s method for functions of multivariate normal random variables

Abstract

It is a well-known fact that if the random vector W converges in distribution to a multivariate normal random variable Σ 1/2 Z, then g(W) converges in distribution to g(Σ 1/2 Z) if g is continuous. In this paper, we develop a general method for deriving bounds on the distributional distance between g(W) and g(Σ 1/2 Z). To illustrate this method, we obtain several bounds for the case that the j-component of W is given byXij , where the Xij are independent. In particular, provided g satisfies certain differentiability and growth rate conditions, we obtain an order n −(p−1)/2 bound, for smooth test functions, if the first p moments of the Xij agree with those of the normal distribution. If p is an even integer and g is an even function, this convergence rate can be improved further to order n −p/2 . We apply these general bounds to some examples concerning asymptotically chi-square, variance-gamma and chi distributed statistics. Primary 60F05.

Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I

“…The O(p −1 ) bound (1.6) is the first faster than O(p −1/2 ) bound for the random sum S n in the literature. The faster convergence rate is a result of the vanishing third moment assumption, and as such complements a number of other 'matching moments' limit theorems that are found in the Stein's method literature, see, for example, [4,12,15,18,21,28]. Theorem 1.3 gives the first bounds in the literature on the rate of convergence of the deterministic sum T n to its asymptotic Laplace distribution.…”

Higher-order Stein kernels for Gaussian approximation

“…Such results have been obtained in dimension one (and for random vectors with independent coordinates) in [20,21]. See also [22] for related results.…”

Statistical Approximating Distributions Under Differential Privacy

“…This suggests that an even stronger notion of convergence is occurring. One possibility is to use several measures of the distance between two random variables X and Y as follows [Gaunt, 2015]. For example, let H be a class of bounded functions that are -times continuously differentiable with bounded derivatives and let G be a class of functions that are -times continuously differentiable with derivatives that are bounded by a specific polynomial.…”

“…For instance, G could consist of the single function z → z 2 that performs squaring. Following Gaunt [2015], define:…”

“…The proof can be found in Appendix A.6 Nonprivate multivariate convergence results of the necessary form are an open area of research for many applications [Gaunt, 2015].…”