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

A. D. Barbour, Nathan Ross, Guangqu Zheng

TL;DR

A general approach to Stein's method for approximating a random process in the path space $D([0,T]\to R^d)$ by a real continuous Gaussian process is developed, deriving a general quantitative Gaussian approximation.

Abstract

We develop a general approach to Stein's method for approximating a random process in the path space $D([0,T]\to R^d)$ by a real continuous Gaussian process. We then use the approach in the context of processes that have a representation as integrals with respect to anunderlying point process, deriving a general quantitative Gaussian approximation. The error bound is expressed in terms of couplings of the original process to processes generated from the reduced Palm measures associated with the point process. As applications, we study certain $\text{GI}/\text{GI}/\infty$ queues in the "heavy traffic" regime.

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

TL;DR

A general approach to Stein's method for approximating a random process in the path space by a real continuous Gaussian process is developed, deriving a general quantitative Gaussian approximation.

Abstract

We develop a general approach to Stein's method for approximating a random process in the path space by a real continuous Gaussian process. We then use the approach in the context of processes that have a representation as integrals with respect to anunderlying point process, deriving a general quantitative Gaussian approximation. The error bound is expressed in terms of couplings of the original process to processes generated from the reduced Palm measures associated with the point process. As applications, we study certain queues in the "heavy traffic" regime.

Paper Structure

This paper contains 10 sections, 13 theorems, 220 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. Test functions
  4. ${\rm M/GI}/\infty$ queue
  5. $\text{GI}/\text{GI}/\infty$ queue
  6. General approximation theorem
  7. Stein's method for Gaussian processes
  8. ${\rm M/GI/\infty}$ queue: Proof of Theorem \ref{['thm:mginfqstn']}
  9. ${\rm GI/GI/\infty}$ queue: Proof of Theorem \ref{['thm:gginfqstn']}
  10. Smoothing result

Key Result

Theorem 1.1

Assume that the convolution $G*\alpha$ defined by $(G*\alpha)(s) := \int_0^s G(s-u)\alpha(du)$, the cumulative intensity $A$ defined by $A(s) := \int_0^s\alpha(du)$, and the distribution function ${\widetilde{G}}$ are all Hölder continuous with exponent $\beta \in (1/2,1]$ and constants $c_{G,\alpha where $x\geqslant0$ is fixed. Now set ${\widehat{J}}_{(t,y)} := J_{t,y}$ for $t>0$ and ${\widehat{J

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 25 more