Workshop MESA - Stein's Method and Applications

Malliavin calculus for marked binomial processes and Chen-Stein method

21 mars 2023, 16:20

55m

Amphi Schwartz (Institut de Mathématiques de Toulouse)

Orateur

Hélène Halconruy (ESILV)

Description

We can observe a clumping phenomenon when counting the number of series of $t$ heads in a sequence of independent coin tosses or the occurrences of a rare word in a DNA sequence. The Chen-Stein method is an efficient tool to limit the approximation error when the law of the number of clusters can be approximated by a Poisson law (possibly compound).
We revisit this method by reducing these two problems to that of a Poisson approximation for functionals of marked binomial processes (MBPs), which are discrete analogues of marked Poisson processes. We then develop stochastic analysis tools and a Malliavin calculus for MBPs. Under this new formalism, we obtain a general criterion - for the distance in total variation - of the Poisson approximation for MBP functionals, in terms of Malliavin operators. In this talk, I will give elements of the Malliavin formalism for MBPs, state the general result of the approximation and illustrate it by applying it to the two situations of interest.

Documents de présentation

halconruy.pdf