Abstract
In this paper we develop a framework for multivariate functional approximation by a suitable Gaussian process via an exchangeable pairs coupling that satisfies a suitable approximate linear regression property, thereby building on work by Barbour (1990) and Kasprzak (2020). We demonstrate the applicability of our results by applying it to joint subgraph counts in an Erd\H{o}s-Renyi random graph model on the one hand and to vectors of weighted, degenerate $U$-processes on the other hand. As a concrete instance of the latter class of examples, we provide a bound for the functional approximation of a vector of success runs of different lengths by a suitable Gaussian process which, even in the situation of just a single run, would be outside the scope of the existing theory.
Highlights
In his seminal paper [64], Charles Stein introduced a method for proving normal approximations and obtained a bound on the speed of convergence to the standard normal distribution
The very recent paper [25] developed a functional analytic approach that provides a substantial extension of the method of exchangeable pairs and, in particular, makes it possible to dispense with the linear regression property in finite-dimensional settings
We look at a Erdos-Renyi random graph with nt vertices, where t denotes the time, and study the distance from the asymptotic distribution of the joint law of the number of edges and the number of twostars
Summary
Consider an Erdos-Renyi random graph G( nt , p) on nt vertices, for t ∈ [0, 1], with a fixed edge probability p. Let Ii,j = Ij,i’s be i.i.d. Bernoulli (p) random variables indicating that edge (i, j) is present in this graph. We consider the following process, representing the re-scaled total number of edges nt − 2 nt − 2. (Ii,j Ij,k + Ii,j Ii,k + Ij,kIi,k). Let Yn(t) = (Tn(t) − ETn(t), Vn(t) − EVn(t)) for t ∈ [0, 1]. P2 and, by an argument similar to that of [54, Section 5], the covariance matrix of (Tn(t) − ETn(t), Vn(t) − EVn(t)) is given by nt n4 p(1 − p). The scaling ensures that the covariances are of the same order in n
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