Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition

Abstract

In this paper we establish a multivariate exchangeable pairs approach within the framework of Stein's method to assess distributional distances to potentially singular multivariate normal distributions. By extending the statistics into a higher-dimensional space, we also propose an embedding method which allows for a normal approximation even when the corresponding statistics of interest do not lend themselves easily to Stein's exchangeable pairs approach. To illustrate the method, we provide the examples of runs on the line as well as double-indexed permutation statistics.Heuristically, (1.1) can be understood as a linear regression condition. If (W, W ′ ) were bivariate normal with correlation ρ, then

Stein's Method and Stochastic Analysis of Rademacher Functionals

“…Then (F, F ′ ) forms an exchangeable pair. In [34] an embedding approach is introduced, which suggests enhancing the pair (F, F ′ ) to a pair of vectors (W, W ′ ) which then ideally satisfy the linearity condition…”

“…Then the results from [34] can be applied to assess the distance of W to a normal distribution with the same covariance matrix. Note that due to the orthogonality of the integrals, the covariance matrix will be zero off the diagonal.…”

“…In particular, this exchangeable pair satisfies a linearity condition in conditional expectation exactly, which allows the normal approximation results from Reinert and Röllin [34] to be applied.…”

Concentration of measures via size-biased couplings

“…, ξ n of n i.i.d Bernoulli(p) random variables with p ∈ (0, 1), given by Y = n i=1 X i where X i = ξ i ξ i+1 · · · ξ i+m−1 , with the periodic convention ξ n+k = ξ k . In [27], the authors develop smooth function bounds for normal approximation for the case of 2-runs. Note that the construction given in Lemma 4.1 for this case is monotone, as for any i, letting…”

Gaussian approximation of suprema of empirical processes

“…We prove an abstract approximation theorem (Theorem 2.1) that leads to results of type (1) in several situations. The proof of the approximation theorem builds on a number of technical tools that are of interest in their own rights: notably, 1) a new coupling inequality for maxima of sums of random vectors (Theorem 4.1), where Stein's method for normal approximation (building here on [7] and originally due to [54,55]) plays an important role (see also [50,44,9]); 2) a deviation inequality for suprema of empirical processes that only requires finite moments of envelope functions (Theorem 5.1), due essentially to the recent work of [4], complemented with a new "local" maximal inequality for the expectation of suprema of empirical processes that extends the work of [59] (Theorem 5.2). We study applications of this approximation theorem to local and series empirical processes arising in nonparametric estimation via kernel and series methods, and demonstrate that our new technique is able to provide the Gaussian approximation for the supremum type statistics under weak regularity conditions, especially concerning the bandwidth and the number of series functions, in those examples.…”