Stein’s method in infinite dimension
The Stein method is a generic method to evaluate distance between probability measures. It is often applied to prove convergence towards Gaussian or Poisson distribution. Its principle is summarized by the following graph.
To compare the probability measures (m) and (m_beta), one constructs a Markov process which is ergodic, whose stationary distribution is (m_beta) and semi-groupe (P_t^beta). By ergodicity, whatever the starting measure (m), at time infinity, one attains (m_beta). Then, one goes back in time controlling the deviation along the reverse path. In this part, properties of the measure (m) are then used. When combined with Malliavin calculus, it is an integration by parts formula which is useful.
We apply this procedure to Brownian approximations: Donsker Theorem, convergence of Poisson process to Brownian motion, linear interpolation of Brownian motion.