  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.
