Stein’s method

Si l'on suivait les voies ferroviaires, qui aurait le pied marin ?

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.

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-group $P_t^{\beta}$. 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. A more detailed introduction to this line of thought for SM is developed in this article

• Donsker theorem, affine interpolation of the Brownian motion.  Beyond the Stein’s method in dimension 1 or $n$, one can investigate what happens for sample-paths of stochastic processes. The first question which comes to mind is that of the rate of convergence of a random walk to the Brownian motion.

A Brownian path and its affine interpolation

The second concerns the convergence of the conveniently renormalized Poisson process towards the same Brownian motion. The new challenges comes mainly from the infinite dimensional spaces we have to deal with.

In this article, we evaluate the convergence rate of the usual random walk to the Brownian motion in some fractional Sobolev spaces. The slides are also available.

• Poisson point processes.  The Poisson point process is the first brick on which the stochastic geometry is built upon. Since we have a very well developed Malliavin calculus for this process, it is rather straightforward to apply the Stein-Malliavin-Dirichlet to this setting. This gives raise to this beautiful paper. We investigate several multi-points transformations of point processes which lead to a Poissonian limit. Once again, a cornerstone of the calculations is the Stein representation formula of the Rubsintein distance as an integral along the path of an Ornstein-Ulhenbeck type process. The magic here is that the whole machinery can also be used to prove convergence of some U-statistics without relying on some artificial hypothesis like having integer-valued marks in the limit.
• Rough paths and Donsker theorem again Rough paths were introduced by Lyons in 1998 to cope with the non-continuity of the Itô map (the map which sends a sample-path of the Brownian motion to the solution of an SDE driven by this path). The so-called enriched Brownian motion is the couple made by the Brownian itself and its Lévy area. It is thus a natural question to enrich the Donsker theorem and see if the adjunction of this new component does change the convergence rate we established earlier. We prove in this article that the rate is unchanged whatever the integrability of the random variables involved in the random walk. How ever, the more integrable they are, the more Hölderian are their sample-paths hence the richer is the convergence.
• More to come, stay tuned.