Note on A. Barbour’s paper on Stein’s method for diffusion approximations

Abstract

In [2] foundations for diffusion approximation via Stein's method are laid. This paper has been cited more than 130 times and is a cornerstone in the area of Stein's method (see, for example, its use in [1] or [7]). A semigroup argument is used in [2] to solve a Stein equation for Gaussian diffusion approximation. We prove that, contrary to the claim in [2], the semigroup considered therein is not strongly continuous on the Banach space of continuous, real-valued functions on D[0, 1] growing slower than a cubic, equipped with an appropriate norm. We also provide a proof of the exact formulation of the solution to the Stein equation of interest, which does not require the aforementioned strong continuity. This shows that the main results of [2] hold true. Definitions and notationBy D = D[0, 1] we will mean the Skorohod space of all the càdlàg functions w : [0, 1] → R. In the sequel · will always denote the supremum norm. By D k f we mean the k-th Fréchet derivative of f and the k-linear norm B is defined to be B =

Stein’s method of exchangeable pairs in multivariate functional approximations

“…We subsequently calculate its infinitesimal generator A n and take it as our Stein operator. Next, we solve the Stein equation A n f = g, using the analysis of [44], and prove several smoothness properties of the solution f n = φ n (g).…”

Stein's method of exchangeable pairs in multivariate functional approximations

“…We subsequently calculate its infinitesimal generator A n and take it as our Stein operator. Next, we solve the Stein equation A n f = g, using the analysis of [KDV17], and prove several smoothness properties of the solution f n = φ n (g).…”

Stein's method, Gaussian processes and Palm measures, with applications to queueing

“….. Following [Barbour, 1990] (see also [Kasprzak, Duncan, and Vollmer, 2017]), for g : D p → R, we define where A := sup w: w =1 |A[w [k] ]| for A a k-linear form, and A[w [k] ] := A [w, w, . .…”