Stein's method for functions of multivariate normal random variables

Abstract

By the continuous mapping theorem, if a sequence of d-dimensional random vectors (Wn)n≥1 converges in distribution to a multivariate normal random variable Σ1/2Z, then the sequence of random variables (g(Wn))n≥1 converges in distribution to g(Σ1/2Z) if g : ℝd → ℝ is continuous. In this paper, we develop Stein's method for the problem of deriving explicit bounds on the distance between g(Wn) and g(Σ1/2Z) with respect to smooth probability metrics. We obtain several bounds for the case that the j-component of Wn is given by Wn,j = 1/√n Σni=1 Xij, 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. These convergence rates are shown to be of optimal order. We apply our general bounds to some examples, which include the distributional approximation of asymptotically chi-square distributed statistics; the approximation of expectations of smooth functions of binomial and Poisson random variables; rates of convergence in the delta method; and a quantitative variance-gamma approximation of the D∗2 statistic for alignment-free sequence comparison in the case of binary sequences.