Malliavin calculus for independent random variables


How to construct a Malliavin gradient on a family of independent variables.

In Stochastic Processes and their Applications, 2019

In the Stein's method, there are three competing approaches to make the core calculations.

  • exchangeable pairs
  • *-biased coupling
  • Malliavin integration by parts

One question (at least for me) is to know whether they are equivalent. Actually, the main drawback of the first two is that they require an adhoc construction at each new situation. On the other hand, Malliavin integration by parts is more universal. We know from the papers of Nourdin, Peccati and others how to use Malliavin IBP when the random variables we handle depend on Poisson or Gaussian processes but it was rather irritiating to see that the simplest situation of all, i.e. the standard central limit theorem for independent random variables, was not accessible through this approach.

As far as the CLT is concerned, The trick consists in constructing an exchangeable pair as follows:

  • Draw $I$ a uniform random variable on $\{1,\cdots,n\}$
  • Exchange $X_I$ with $X'$ an independent copy of $X$
  • Then the pair $(\sum X_i, \sum X_i - X_I +X')$ is exchangeable

and the rest of the proof can follow. The main motivation of this paper was to find a good definition of a Malliavin gradient so that the CLT was provable via an integration by parts. Take $$ D_iF = F-E[F \, |\, X_j, j\neq i]$$ and you are done!

L. Decreusefond
Professor of Probability