Functional approximations via Stein’s method of exchangeable pairs

Stein’s method of exchangeable pairs in multivariate functional approximations

“…Theorem 6.4 estimates the distance between the law of (1.2) and that of a continuous Gaussian process. These results extend the result of [42] bounding the distance between the distribution of the edge counts and a univariate Gaussian process. As a corollary to our results, we immediately obtain weak convergence of the law of (1.2) in the Skorokhod and uniform topologies on the Skorokhod space to that of the continuous Gaussian process.…”

“…Approximations by laws of stochastic processes using Stein's method have been studied in [3,5,19,20,63] and recently in [7,10,21,[41][42][43]. These references can be divided into three groups.…”

“…The ones belonging to the first group, containing [3,5,[41][42][43], all use, adapt and extend the setup of [3]. Therein, the author studied the rate of convergence in the celebrated functional central limit theorem, also called Donsker's theorem.…”

“…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. In [42] the method was applied to the study of functional limit results and approximations by univariate Gaussian processes, using the setup of [56,65] and [3].…”

“…This strategy, in a quantified fashion, is exemplified by the application to the rruns process, discussed in Subsection 5.6. We stress that, even in the case of just one r-run process, the results about univariate functional approximations via exchangeable pairs from [42] would not be sufficient to obtain a Gaussian approximation. Thus, in this example, the multidimensionality of our approach proves to be absolutely vital.…”