Diffusive limits of Lipschitz functionals of Poisson measures

Abstract

Continuous Time Markov Chains, Hawkes processes and many other interesting processes can be described as solution of stochastic differential equations driven by Poisson measures. Previous works, using the Stein's method, give the convergence rate of a sequence of renormalized Poisson measures towards the Brownian motion in several distances, constructed on the model of the Kantorovitch-Rubinstein (or Wasserstein-1) distance. We show that many operations (like time change, convolution) on continuous functions are Lipschitz continuous to extend these quantified convergences to diffuse limits of Markov processes and long-time behavior of Hawkes processes.

References in corpus (9)

• Large graph limit for an SIR process in random network with heterogeneous connectivity
• Heavy traffic analysis for EDF queues with reneging
• Stein's method for normal approximation in Wasserstein distances with application to the multivariate Central Limit Theorem
• A functional central limit theorem for the M/GI/$\infty$ queue
• Gaussian approximation of nonlinear Hawkes processes
• Stein's method for steady-state diffusion approximations of $M/Ph/n+M$ systems
• A multivariate central limit theorem for Lipschitz and smooth test functions
• Diffusion approximations via Stein's method and time changes
• Donsker's theorem in {Wasserstein}-1 distance

Cited by in corpus (3)

• Stein's method, Gaussian processes and Palm measures, with applications to queueing
• Rate of Convergence in the Functional Central Limit Theorem for Stable Processes
• Gaussian Approximation and Moderate Deviations of Poisson Shot Noises with Application to Compound Generalized Hawkes Processes